Your FREE Actuarial Exam P (Probability) Practice Test 2026 – 260+ Q&A

Correct answer: The set of all possible outcomes of the experiment

The correct choice is that the sample space is the set of all possible outcomes of the experiment. By definition, the sample space (often denoted S or Omega) lists every outcome the experiment could produce, and any event is a subset of it. It is not a single outcome, a partial collection, nor the most likely outcome, since it must contain every possibility.

Correct answer: 1

The correct answer is 1. The probability axioms (Kolmogorov's axioms) require that P(S) = 1 for the sample space S, meaning some outcome in S is certain to occur. A value of 0 would mean the experiment yields nothing, 0.5 has no basis, and probabilities can never exceed 1 or be infinite.

Correct answer: 0 <= P(A) <= 1

The correct statement is 0 <= P(A) <= 1. A core axiom of probability is nonnegativity combined with the bound that no event can be more certain than the whole sample space, so every probability lies between 0 and 1 inclusive. Negative probabilities are disallowed, the strict inequalities wrongly exclude impossible and certain events, and probabilities cannot exceed 1.

Correct answer: 0

The correct value is 0. Mutually exclusive (disjoint) events cannot happen at the same time, so their intersection is empty and P(A and B) = 0. The product P(A)P(B) would apply only to independent events, the sum gives the probability of their union, and 1 has no justification here.

Correct answer: P(A) + P(B) - P(A and B)

The correct expression is P(A) + P(B) - P(A and B). The general addition rule subtracts the overlap once because outcomes in both events are otherwise counted twice. Simply adding P(A) and P(B) double-counts the intersection, the product applies to independence not unions, and adding the intersection inflates the result further.

Correct answer: 0.42

The correct probability is 0.42. Using the addition rule, P(auto or home) = 0.30 + 0.20 - 0.08 = 0.42, where subtracting the 0.08 overlap prevents double-counting those who file both. Adding without subtracting gives 0.50, and the other values come from incorrect arithmetic.

Correct answer: 0.3

The correct answer is 0.3. For independent events the multiplication rule gives P(A and B) = P(A)P(B) = 0.6 x 0.5 = 0.3. Adding the probabilities produces 1.1 which is invalid, while 0.5 and 0.6 simply restate the inputs.

Correct answer: P(A and B) = P(A)P(B)

The defining condition is P(A and B) = P(A)P(B). Independence means the occurrence of one event does not change the probability of the other, which is captured exactly by the joint probability factoring into the product. An intersection of 0 describes mutually exclusive events, the additive relation describes disjoint events, and equal marginal probabilities are unrelated to independence.

Correct answer: $\frac{1}{8}$

The correct probability is 1/8. Because the flips are independent, the multiplication rule gives $\frac{1}{2}\frac{1}{2}\frac{1}{2}$ = 1/8. The value $\frac{3}{8}$ confuses this with exactly one specific arrangement count, $\frac{1}{2}$ ignores the multiple flips, and $\frac{1}{6}$ has no basis.

Correct answer: P(A and B) / P(B)

The correct definition is P(A and B) / P(B). Conditional probability rescales the joint probability of A and B by the probability of the conditioning event B, reflecting that we now restrict attention to outcomes where B occurred. Dividing by P(A) reverses the roles, the ratio of marginals ignores the joint behavior, and multiplying does not yield a conditional probability.

Correct answer: 0.10

The correct probability is 0.10. Using P(dog and cat) = P(dog) x P(cat given dog) = 0.40 x 0.25 = 0.10. Adding the percentages gives 0.65, dividing gives 0.625, and 0.15 results from incorrect subtraction; only the multiplication rule with the conditional probability is valid here.

Correct answer: $\frac{7}{15}$

The correct probability is 7/15. Without replacement, P(both red) = $\frac{7}{10}\frac{6}{9}$ = $\frac{42}{90}$ = $\frac{7}{15}$, where the second factor reflects that one red marble is already removed. The value $\frac{49}{100}$ wrongly assumes replacement, $\frac{21}{45}$ is unsimplified and incorrect, and $\frac{3}{10}$ ignores the joint draw.

Correct answer: 60

The correct count is 60. Because order matters and letters are not repeated, this is a permutation: 5 x 4 x 3 = 60. The value 10 is the number of combinations (order ignored), 15 has no basis, and 125 comes from allowing repeats (5 cubed).

Correct answer: 56

The correct count is 56. Since order does not matter, this is a combination: 8 choose 3 = (8 x 7 x 6)/(3 x 2 x 1) = 56. The value 336 is the permutation count where order matters, 24 and 512 come from incorrect setups.

Correct answer: (4 choose 1)(6 choose 2) / (10 choose 3)

The correct expression is (4 choose 1)(6 choose 2) / (10 choose 3). The numerator counts ways to pick exactly one of the 4 defectives and two of the 6 working units, while the denominator counts all ways to choose 3 of the 10 total components. The other denominators use wrong totals, and omitting the (6 choose 2) factor undercounts the favorable selections.

Correct answer: 13

The correct count is 13. The elements in A but not B equal the total in A minus those in the overlap: 18 - 5 = 13. The value 18 ignores the shared region, 7 mistakenly uses B's count, and 23 incorrectly adds rather than subtracts.

Correct answer: 35%

The correct percentage is 35%. By the inclusion-exclusion (addition) rule, P(both) = P(X) + P(Y) - P(at least one) = 0.70 + 0.55 - 0.90 = 0.35. The other values come from adding instead of subtracting or from arithmetic errors in applying the overlap relationship.

Correct answer: 0.032

The correct probability is 0.032. The law of total probability weights each machine's defect rate by its share of output: (0.50)(0.02) + (0.30)(0.04) + (0.20)(0.05) = 0.010 + 0.012 + 0.010 = 0.032. Simply averaging the rates or summing them gives the other values, which ignore the production weights.

Correct answer: (0.99)(0.01) / [(0.99)(0.01) + (0.05)(0.99)]

The correct expression is (0.99)(0.01) / [(0.99)(0.01) + (0.05)(0.99)]. Bayes' Theorem divides the probability of a true positive (sensitivity times prevalence) by the total probability of a positive result, which adds true positives and false positives, where false positives equal (1 - specificity) times the healthy proportion, namely (0.05)(0.99). The other options misuse specificity in place of the false-positive rate or attach the wrong sensitivity factor to the numerator.

Correct answer: The complement of (A union B) equals (A complement) intersect (B complement)

The correct statement is that the complement of (A union B) equals the intersection of the complements, (A complement) intersect (B complement). De Morgan's law says 'not (A or B)' is the same as 'not A and not B,' since an outcome avoids the union only when it lies outside both sets.

Correct answer: P(A)+P(B)+P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C)

The answer adds the single probabilities, subtracts all three pairwise intersections, then adds back the triple intersection: P(A)+P(B)+P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C). This corrects for elements counted multiple times across the overlapping sets.

Correct answer: 0.6

The probability is 0.6. Odds in favor of 3 to 2 mean 3 favorable outcomes out of 3+2 = 5 total equally weighted parts, so the probability equals $\frac{3}{5}$ = 0.6.

Correct answer: 1 to 4

The odds against are 1 to 4. With probability 0.8 of occurring and 0.2 of not occurring, the odds against equal the ratio of failure to success, 0.2 to 0.8, which simplifies to 1 to 4.

Correct answer: 0.2

The probability is 0.2. For a uniform choice on a segment, probability equals favorable length over total length; the points within 2 of the left endpoint form a sub-segment of length 2, so the probability is $\frac{2}{10}$ = 0.2.

Correct answer: $\frac{\pi}{4}$

The probability is $\frac{\pi}{4}$. For uniform placement, probability equals the circle's area over the square's area: the circle area is pi times 2 squared = 4pi, and the square area is 16, giving $\frac{4\pi}{16}$ = $\frac{\pi}{4}$.

Correct answer: 1296

There are 1296 outcomes. By the multiplication (counting) principle, each of the four rolls has 6 possibilities independently, so the total is 6 raised to the 4th power, which equals 1296.

Correct answer: 58500

There are 58500 passwords. The two distinct ordered letters give 26 times 25 = 650 arrangements, the two distinct ordered digits give 10 times 9 = 90 arrangements, and by the multiplication principle 650 times 90 = 58500.

Correct answer: 120

There are 120 arrangements. The number of orderings of 5 distinct objects is 5 factorial = 5 times 4 times 3 times 2 times 1 = 120.

Correct answer: 84

There are 84 triangles. A triangle is an unordered choice of 3 points from 9, which is the combination $\binom{9}{3}$ = (9 times 8 times 7)/(3 times 2 times 1) = 84.

Correct answer: 125970

There are 125970 ways. Choosing the group of 12 from 20 is the combination $\binom{20}{12}$, which equals $\binom{20}{8}$ = 125970; selecting the 12 automatically determines the remaining 8.

Correct answer: 35

There are 35 arrangements. With repeated identical items, the count is 7 factorial divided by (4 factorial times 3 factorial) = 5040/(24 times 6) = 35, equivalent to choosing which 4 of the 7 positions hold red flags.

Correct answer: $\frac{5}{36}$

The probability is 5/36. Among the 36 equally likely ordered outcomes, the pairs summing to 8 are (2,6),(3,5),(4,4),(5,3),(6,2) -- exactly 5 outcomes -- giving 5/36.

Correct answer: $\frac{5}{16}$

The probability is 5/16. The number of ways to place 3 heads among 5 flips is $\binom{5}{3}$ = 10, and each specific sequence has probability ($\frac{1}{2}$) to the 5th = $\frac{1}{32}$, so $\frac{10}{32}$ = 5/16.

Correct answer: $\frac{1}{6}$

The probability is 1/6. The ways to draw 3 of one color are $\binom{5}{3}$ = 10 for red plus 10 for blue = 20, and the total ways to draw 3 from 10 is $\binom{10}{3}$ = 120, giving $\frac{20}{120}$ = 1/6.

Correct answer: $\frac{7}{15}$

The probability is 7/15. Matching pairs are $\binom{6}{2}$ = 15 black plus $\binom{4}{2}$ = 6 white = 21, out of $\binom{10}{2}$ = 45 total pairs, so $\frac{21}{45}$ = 7/15.

Correct answer: 0.5

The probability is 0.5. Exactly one occurs means (A and not B) or (B and not A): 0.5 times 0.4 = 0.20 plus 0.5 times 0.6 = 0.30, summing to 0.50.

Correct answer: A complement and B complement are also independent

The correct statement is that A complement and B complement are also independent. If A and B are independent, then their complements are independent as well, since P(A' and B') = P(A')P(B') follows directly from the independence of A and B.

Correct answer: 0.75

The probability is 0.75. For mutually exclusive events the intersection probability is 0, so P(A union B) = P(A) + P(B) = 0.30 + 0.45 = 0.75.

Correct answer: No, because mutual exclusivity forces P(A and B) = 0 while independence requires P(A)P(B) > 0

No: mutual exclusivity forces P(A and B) = 0, while independence with positive probabilities requires P(A and B) = P(A)P(B) > 0. These two conditions cannot both hold, so positively probable mutually exclusive events are necessarily dependent.

Correct answer: $\frac{1}{3}$

The probability is 1/3. There are 12 face cards and 4 of them are Kings, so the conditional probability is $\frac{4}{12}$ = 1/3.

Correct answer: $\frac{1}{3}$

The probability is 1/3. The equally likely outcomes given at least one girl are GG, GB, BG (BB excluded), and only GG has both girls, giving 1/3.

Correct answer: $\frac{13}{52}\frac{12}{51}\frac{11}{50}$

The probability is $\frac{13}{52}\frac{12}{51}\frac{11}{50}$. Using the multiplication rule for dependent draws, each successive heart reduces both the hearts remaining and the total cards remaining, producing this product of conditional probabilities.

Correct answer: $\frac{7}{30}$

The probability is 7/30. The first draw must be silver ($\frac{7}{10}$) and the second gold ($\frac{3}{9}$), and $\frac{7}{10}\frac{3}{9}$ = $\frac{21}{90}$ = 7/30.

Correct answer: P(A complement) = 1

The guaranteed result is P(A complement) = 1. The complement rule states P(A) + P(A complement) = 1, so if P(A) = 0 then P(A complement) = 1.

Correct answer: P(A) <= P(B)

The required inequality is P(A) <= P(B). Monotonicity of probability states that if A is contained in B, then every outcome in A is also in B, so P(A) cannot exceed P(B).

Correct answer: P(A union B) <= P(A) + P(B)

The always-true relationship is P(A union B) <= P(A) + P(B). Because P(A union B) = P(A) + P(B) - P(A and B) and the intersection probability is nonnegative, the union probability is at most the sum of the individual probabilities.

Correct answer: $\frac{1}{2}$

The probability is 1/2. There are 10 beads, 5 of which are blue, so 5 are not blue, giving $\frac{5}{10}$ = $\frac{1}{2}$ by the complement of drawing blue.

Correct answer: $\frac{11}{36}$

The probability is 11/36. Using the complement, the probability that neither die is a 6 is $\frac{5}{6}\frac{5}{6}$ = $\frac{25}{36}$, so at least one 6 has probability 1 - $\frac{25}{36}$ = 11/36.

Correct answer: $\frac{1}{16}$

The probability is 1/16. The first friend's choice is free; the second matches with probability $\frac{1}{4}$ and the third with probability $\frac{1}{4}$, so $\frac{1}{4}\frac{1}{4}$ = 1/16.

Correct answer: (365 times 364 times 363 times 362)/$365^{4}$

The expression is (365 times 364 times 363 times 362)/$365^{4}$. The numerator counts ordered selections of 4 distinct birthdays and the denominator counts all equally likely birthday assignments, so their ratio is the probability of no shared birthday.

Correct answer: $\frac{1}{64}$

The probability is 1/64. Each guess is correct with probability $\frac{1}{4}$, and since the guesses are independent, the probability of all three correct is ($\frac{1}{4}$) cubed = 1/64.

Correct answer: 0.512

The probability is 0.512. Each buyer declines the warranty with probability 0.8, and by independence the probability all three decline is 0.8 cubed = 0.512.

Correct answer: (0.02)(0.95) + (0.98)(0.10)

The denominator is (0.02)(0.95) + (0.98)(0.10). By the law of total probability, the overall probability of a positive test sums the true positives among the diseased and the false positives (1 - 0.90 = 0.10) among the healthy.

Correct answer: 0.04/0.058

The probability is 0.04/0.058. The standard-risk claim probability is (0.40)(0.10) = 0.04, and the total claim probability is 0.04 + (0.60)(0.03) = 0.04 + 0.018 = 0.058, so Bayes' Theorem gives 0.04/0.058.

Correct answer: 0.34

The probability is 0.34. With P(C) = 1 - 0.5 - 0.3 = 0.2, the law of total probability gives P(E) = (0.5)(0.2) + (0.3)(0.4) + (0.2)(0.6) = 0.10 + 0.12 + 0.12 = 0.34.

Correct answer: 60

The number is 60. Smartphone owners who do not own a tablet equal total smartphone owners minus those owning both: 120 - 60 = 60.

Correct answer: 0.4

The probability is 0.4. For a uniform choice on a line segment, the probability of landing in a subinterval equals the subinterval length divided by the total length: (7-3)/(10-0) = $\frac{4}{10}$ = 0.4. Geometric probability uses length ratios for one-dimensional sample spaces.

Correct answer: 252

There are 252 ways. Because the teams are labeled, choosing which 5 files go to Team Day determines the rest, giving the combination $\binom{10}{5}$ = 10!/(5!5!) = 252. No additional division by 2 is needed since the labels make the two groups distinguishable.

Correct answer: 0.999

The probability is 0.999. The complement of at least one functioning is none functioning, which by independence is (0.1)(0.1)(0.1) = 0.001, so the answer is 1 - 0.001 = 0.999. The complement rule turns an at-least-one question into a single product.

Correct answer: The events have no common outcomes and together cover the entire sample space, so their probabilities sum to 1

A mutually exclusive and exhaustive partition has no shared outcomes and its union is the whole sample space, forcing P(E1)+P(E2)+P(E3) = 1. Mutually exclusive means disjoint events; exhaustive means they collectively exhaust every possible outcome.

Correct answer: $\frac{15}{28}$

The probability is 15/28. A mixed pair occurs as red-then-blue or blue-then-red: $\frac{5}{8}\frac{3}{7}$ + $\frac{3}{8}\frac{5}{7}$ = $\frac{15}{56}$ + $\frac{15}{56}$ = $\frac{30}{56}$ = 15/28. Summing the two ordered sequences captures both ways to obtain one sock of each color.

Correct answer: Yes, because P(A)P(B) = 0.22 equals P(A and B)

Yes, A and B are independent. Independence holds exactly when P(A and B) = P(A)P(B). Here P(A)P(B) = 0.55 times 0.40 = 0.22, which matches the given P(A and B) = 0.22, so the events are independent.

Correct answer: $\frac{1}{4}$

The answer is 1/4. Among the 12 face cards, exactly 3 are diamonds (jack, queen, king of diamonds), so P(A given B) = $\frac{3}{12}$ = 1/4. Equivalently, the suit is independent of being a face card.

Correct answer: 0.55

The answer is 0.55. Exactly one equals P(P) + P(Q) minus twice the overlap: 0.35 + 0.50 minus 2 times 0.15 = 0.85 minus 0.30 = 0.55.

Correct answer: 0.0904

The answer is 0.0904. By the law of total probability, P(positive) = 0.06 times 0.88 + 0.94 times 0.04 = 0.0528 + 0.0376 = 0.0904.

Correct answer: 0.0975

The answer is 0.0975. The signal succeeds only if both work: 0.95 times 0.95 = 0.9025. The failure probability is the complement, 1 minus 0.9025 = 0.0975.

Correct answer: 20

The answer is 20. Because order does not matter (the letters are forced into alphabetical order), this is a combination: 6 choose 3 = 6!/(3!3!) = 20.

Correct answer: F(x) must integrate to 1 over the support and be nonnegative everywhere

The defining requirements are that f(x) integrate to 1 over the support and be nonnegative everywhere. A valid probability density function must never be negative and must enclose a total area of 1, since total probability is certain. A density value can exceed 1, the probability that a continuous variable equals an exact point is 0 rather than f(x), and a density need not be decreasing.

Correct answer: The probability that X is less than or equal to x

The cumulative distribution function gives the probability that X is less than or equal to x, written F(x) = P(X <= x). It accumulates probability from the lower tail up to the point x. The probability of exceeding x is the complement 1 - F(x), the density is the derivative of F rather than F itself, and F is not an expected value.

Correct answer: By taking the derivative of F(x) with respect to x

The density is obtained by taking the derivative of F(x) with respect to x, since f(x) = F'(x) for a continuous variable. The cumulative distribution function accumulates the density, so differentiation reverses that accumulation. Integrating F gives no meaningful density, the complement 1 - F is the survival function, and squaring has no basis.

Correct answer: 0.7

The expected value is 0.7. Computing the probability-weighted sum gives (0)(0.5) + (1)(0.3) + (2)(0.2) = 0 + 0.3 + 0.4 = 0.7. The value 1.0 is the simple average of the outcomes ignoring probabilities, while 0.5 and 1.5 come from incorrect weighting.

Correct answer: $E(X^{2}) - [E(X)]^2$

The correct identity is $E(X^{2}) - [E(X)]^2$. Variance is the expected squared deviation from the mean, which algebraically simplifies to the second moment minus the square of the first moment. Reversing the terms can give a negative result, subtracting E(X) instead of its square is dimensionally wrong, and adding the terms does not yield variance.

Correct answer: 0.25

The coefficient of variation is 0.25. It is defined as the standard deviation divided by the mean, so 50 divided by 200 equals 0.25. The value 4.0 inverts the ratio, 250 adds the two figures, and 0.50 results from an arithmetic error; only the ratio of standard deviation to mean is correct.

Correct answer: Take the first derivative of M(t) and evaluate at t = 0

The expected value is found by taking the first derivative of M(t) and evaluating at t = 0, giving M'(0) = E(X). The moment generating function produces moments through successive derivatives at the origin. The second derivative at 0 gives the second moment E($X^{2}$), evaluating at t = 1 has no general meaning, and integration does not recover a moment.

Correct answer: The median

That value is the median. The median splits a distribution so that half the probability lies below it, which is exactly the condition F(m) = 0.5. The mode is the most likely value where the density peaks, the mean is the probability-weighted average, and the variance measures spread, none of which is defined by F equaling 0.5.

Correct answer: The value of x where the density f(x) attains its maximum

The mode is the value of x where the density f(x) attains its maximum, representing the most likely region of outcomes. The point where the cumulative distribution equals one half is the median, the probability-weighted average is the mean, and a density crossing zero does not define the mode.

Correct answer: The 90th percentile

That value is the 90th percentile. A percentile is the point below which a stated proportion of the distribution lies, so c with P(X <= c) = 0.90 marks the 90th percentile. The 10th percentile would correspond to a cumulative probability of 0.10, the mean is an average rather than a quantile, and the interquartile range measures spread.

Correct answer: Np

The expected number of successes is np. The binomial mean multiplies the number of trials by the per-trial success probability, reflecting the average count over n independent Bernoulli trials. The expression np(1-p) is the variance, while p/n and n/p invert or misplace the factors.

Correct answer: (8 choose 2)$(0.25)^{2} (0.75)$^6

The correct expression is (8 choose 2)$(0.25)^{2} (0.75)$^6. The binomial probability multiplies the number of ways to place 2 successes among 8 trials by the success probability squared and the failure probability raised to the remaining 6 trials. Swapping the exponents misassigns successes and failures, omitting the combination undercounts arrangements, and using exponent 8 exceeds the number of trials.

Correct answer: The mean and variance are both equal to the rate parameter lambda

The correct statement is that the mean and variance are both equal to the rate parameter lambda. A distinctive feature of the Poisson distribution is the equality of its mean and variance. The other options incorrectly square lambda or scale it, which would break this defining equidispersion property.

Correct answer: $e^{-3}$

The probability is $e^{-3}$. The Poisson probability of k events is $e^{-\lambda}$ $\lambda^{k}$ / k!, so for k = 0 and lambda = 3 it reduces to $e^{-3}$ since $\lambda^{0}$ and 0! both equal 1. The value 3 $e^{-3}$ is the probability of exactly one arrival, 1 - $e^{-3}$ is the probability of at least one, and $e^{-\frac{1}{3}}$ uses the wrong rate.

Correct answer: 1/p

The expected value is 1/p. For the geometric distribution counting trials until the first success, the mean number of trials is the reciprocal of the success probability. The value p is the per-trial probability itself, 1/$p^{2}$ confuses it with a second-moment quantity, and (1-p)/p is the mean when counting failures before the first success rather than total trials.

Correct answer: $(0.8)^{3} (0.2)$

The probability is $(0.8)^{3} (0.2)$. The geometric model requires the first three calls to fail, each with probability 0.8, followed by a success on the fourth with probability 0.2. Swapping the roles of success and failure misassigns the probabilities, using a fourth power of 0.8 adds an extra failure, and $(0.2)^{4}$ ignores the required failures.

Correct answer: Sampling a fixed number of items without replacement from a finite population of two types

The hypergeometric distribution models sampling a fixed number of items without replacement from a finite population of two types, such as defective and non-defective. Because draws are made without replacement, the trials are dependent. Independent trials with replacement describe the binomial, rare events over time describe the Poisson, and trials until the first success describe the geometric.

Correct answer: (5 choose 2)(7 choose 2) / (12 choose 4)

The correct expression is (5 choose 2)(7 choose 2) / (12 choose 4). The hypergeometric formula chooses 2 of the 5 red balls and 2 of the 7 green balls to fill the four-ball sample, divided by the number of ways to choose any 4 of the 12 balls. The other options use the wrong green count, the wrong total in the denominator, or omit the green selection factor.

Correct answer: The number of independent trials needed to achieve a fixed number of successes

The negative binomial models the number of independent trials needed to achieve a fixed number of successes, generalizing the geometric distribution to more than one required success. Counting successes in a fixed number of trials is the binomial, events in a continuous interval is the Poisson, and a proportion is not a count distribution.

Correct answer: Mean 0 and standard deviation 1

The standard normal distribution has mean 0 and standard deviation 1. Standardizing any normal variable shifts it to be centered at 0 and scaled to unit spread, which is why its tables are universally tabulated. The other options assign nonzero means or zero spread, neither of which describes the standard normal.

Correct answer: 2.0

The z-score is 2.0. Standardizing subtracts the mean and divides by the standard deviation, so (660 - 500) divided by 80 equals 160 divided by 80, which is 2.0. The value 1.6 forgets to divide by the standard deviation properly, 0.5 inverts the ratio, and -2.0 has the wrong sign since 660 lies above the mean.

Correct answer: 68%

Approximately 68% lies within one standard deviation of the mean. Under the empirical rule for the normal distribution, about 68% of probability falls within one standard deviation, about 95% within two, and about 99.7% within three. The value 50% understates the central concentration, while 95% and 99.7% correspond to two and three standard deviations.

Correct answer: 25

The variance is 25. For the exponential distribution the variance equals the square of the mean, so with a mean of 5 the variance is 5 squared, which is 25. The value 5 confuses the variance with the mean, 2.5 halves the mean, and 0.2 is the rate parameter rather than the variance.

Correct answer: The memoryless property

The exponential distribution is characterized by the memoryless property, meaning the probability of waiting an additional amount of time does not depend on how much time has already elapsed. This is why it models lifetimes without aging. Symmetry describes the normal distribution, additivity is not a defining trait here, and the exponential has unbounded support.

Correct answer: The same as a brand-new component surviving 500 hours

The probability is the same as a brand-new component surviving 500 hours. The memoryless property of the exponential distribution means the past survival time provides no information about future survival, so a used component behaves like a new one. It is neither more nor less likely to survive based on age, and exceeding the mean does not force failure.

Correct answer: The exponential distribution

The gamma distribution reduces to the exponential distribution when its shape parameter equals 1. A gamma variable can be seen as a sum of independent exponential variables, so a single one recovers the exponential. The normal, binomial, and uniform distributions are not special cases of the gamma at shape parameter 1.

Correct answer: The interval from 0 to 1

The beta distribution is defined over the interval from 0 to 1, making it a natural model for proportions and probabilities. Its bounded support distinguishes it from distributions on all real numbers like the normal, on the nonnegative reals like the exponential, or on a discrete integer range like the binomial.

Correct answer: 6

The expected value is 6. For a continuous uniform distribution the mean is the midpoint of the interval, equal to (2 + 10) divided by 2, which is 6. The values 4 and 8 are interior points but not the center, and 5 is the half-width of the interval rather than its midpoint.

Correct answer: The loss is the full amount of damage, while the payment is what the insurer actually pays after policy provisions

The key distinction is that the loss is the full amount of damage, while the payment is what the insurer actually pays after policy provisions such as deductibles, limits, and coinsurance are applied. The payment is generally smaller than or equal to the loss, not larger, and it reflects deductibles rather than ignoring them, so the two are not always equal.

Correct answer: X - 500

The payment equals X - 500. Under an ordinary deductible, the insurer pays the loss amount reduced by the deductible whenever the loss exceeds it, so the insured absorbs the first 500. Paying the full X ignores the deductible, paying a flat 500 confuses the deductible with the payment, and 500 - X is negative for large losses and has no meaning.

Correct answer: 1600

The insurer pays 1600. Under 80% coinsurance, the insurer covers 80% of the eligible loss, so 0.80 times 2000 equals 1600, with the insured responsible for the remaining 20%. Paying the full 2000 ignores the coinsurance, 400 applies only 20%, and 2500 incorrectly grosses up the amount.

Correct answer: 10,000

The insurer pays 10,000. A benefit limit caps the maximum payment, so any loss above the limit results in a payment equal to the limit itself, leaving the insured to absorb the excess. Paying 13,000 ignores the cap, 3,000 is the uncovered excess rather than the payment, and 11,500 has no basis.

Correct answer: $\frac{1}{9}$

The value of c is 1/9. The density must integrate to 1, so the integral of c $x^{2}$ from 0 to 3 equals c times ($\frac{27}{3}$) = 9c, and setting 9c = 1 gives c = 1/9.

Correct answer: 1

The value of c is 1. The probabilities must sum to 1, and the geometric series of 1/$2^{k}$ for k starting at 1 sums to 1, so c times 1 = 1, giving c = 1.

Correct answer: $e^{-2}$

The probability is $e^{-2}$. Integrating the density $e^{-x}$ from 2 to infinity gives $e^{-2}$, which is the survival probability beyond x = 2.

Correct answer: 40

The answer is 40. Using $Var(X) = E(X^{2}) - (E(X))^2$, we get $E(X^{2}) = Var(X) + (E(X))^2 = 4 + 36 = 40$.

Correct answer: 0

The answer is 0. By linearity of expectation, E(Y) = -2 E(X) + 10 = -2(5) + 10 = 0.

Correct answer: 12

The standard deviation of Y is 12. The constant 2 does not affect spread, and multiplying by 4 scales the standard deviation by the absolute value of 4, giving 4 times 3 = 12.

Correct answer: It is positively (right) skewed

A positive third central moment indicates the distribution is positively (right) skewed, meaning it has a longer tail extending to the right of the mean.

Correct answer: The heaviness of the tails and peakedness

Kurtosis measures the heaviness of the tails and peakedness of a distribution relative to the normal distribution; higher kurtosis indicates more extreme outliers.

Correct answer: 24

The variance is 24. The mean is 0.4(10) + 0.6(20) = 16, and E($X^{2}$) is 0.4(100) + 0.6(400) = 280, so Var(X) = 280 - 256 = 24.

Correct answer: A constant equal to 3

This describes a constant equal to 3. The moment generating function of a degenerate random variable that always equals a constant c is $e^{ct}$, so here c = 3.

Correct answer: The second derivative of M(t) evaluated at t = 0

The second moment E($X^{2}$) is the second derivative of M(t) evaluated at t = 0. In general the nth moment equals the nth derivative of the moment generating function at zero.

Correct answer: 4

The expected value is 4. This is an exponential density with rate $\frac{1}{4}$, so its mean equals the reciprocal of the rate, which is 4.

Correct answer: 100

The expected payment per loss exceeding the deductible is 100. By the memoryless property of the exponential distribution, the excess over the deductible has the same exponential distribution with mean 100.

Correct answer: 5

The mode is 5. For a symmetric binomial with n = 10 and p = 0.5, the most probable outcome is at n times p = 5.

Correct answer: 0.21

The variance is 0.21. For a Bernoulli variable the variance equals p times (1 - p) = 0.3 times 0.7 = 0.21.

Correct answer: 3.5

The expected value is 3.5. For a discrete uniform on integers 1 through n, the mean is (n + 1)/2, which for n = 6 equals 3.5.

Correct answer: 3

The approximate standard deviation is 3. A Poisson variable has variance equal to its mean of 9, so the standard deviation is $\sqrt{9}$, which is 3.

Correct answer: 0.5

The probability is 0.5. The standard normal distribution is symmetric about 0, so exactly half of its area lies above 0.

Correct answer: 850

The loss value is 850. Converting back from the z-score gives x = mean + z times standard deviation = 1000 + (-1.5)(100) = 850.

Correct answer: $\frac{1}{2}$

The probability is 1/2. The interval above 7 has length 3 out of the total length 6, so the probability is $\frac{3}{6}$ = 1/2.

Correct answer: Theta times ln 2

The median equals theta times ln 2. Setting the cumulative distribution 1 - $e^{-x/\theta}$ equal to 0.5 and solving gives x = theta times the natural log of 2.

Correct answer: $\frac{1}{16}$

The probability is 1/16. The cumulative distribution is $x^{4}$, so F(0.5) = $(0.5)^{4}$ = 1/16.

Correct answer: 0.6

The probability is 0.6. The insurer pays the full limit whenever the loss is at least 8000, which for a uniform variable on 0 to 20000 has probability (20000 - 8000)/20000 = 0.6.

Correct answer: $\frac{1}{4}$

At most $\frac{1}{4}$ lies beyond 2 standard deviations. Chebyshev's inequality bounds this probability by 1 divided by k squared, which for k = 2 gives $\frac{1}{4}$, regardless of the specific moments.

Correct answer: $\frac{1}{4}$

The probability is 1/4. The survival function is $\left(\frac{10}{10 + x}\right)^{2}$, and at x = 10 this equals $\left(\frac{10}{20}\right)^{2} = \left(\frac{1}{2}\right)^{2} = \frac{1}{4}$.

Correct answer: $\frac{3}{8}$

The probability is 3/8. There are 3 ways to get exactly 2 heads out of 3 tosses, each with probability $\frac{1}{8}$, giving 3/8.

Correct answer: A normal distribution

For a lognormal random variable, the natural log of X follows a normal distribution; this is the defining property used to model positively skewed quantities like claim sizes.

Correct answer: 6

The expected value is 6. This is a gamma moment generating function with shape 3 and scale 2, so the mean is shape times scale = 3 times 2 = 6.

Correct answer: 400

The expected reimbursement is 400. Multiplying every loss by a constant factor scales the expectation, so E(0.8 X) = 0.8 times 500 = 400.

Correct answer: 5

The median is 5. For any distribution symmetric about a point, that point of symmetry is the median because half the probability lies on each side.

Correct answer: 2

The expected number of failures is 2. For this parameterization the mean equals r times (1 - p) divided by p = 2 times 0.5 divided by 0.5 = 2.

Correct answer: $\frac{1}{3}$

The expected value is 1/3. Integrating x times 2(1 - x) from 0 to 1 gives 2 times ($\frac{1}{2}$ - $\frac{1}{3}$) = 2 times $\frac{1}{6}$ = 1/3.

Correct answer: 2.5

The variance is 2.5. A defining property of the Poisson distribution is that its variance equals its mean, here 2.5.

Correct answer: $(b - a)^{2}$ divided by 12

The variance of a continuous uniform distribution on a to b is (b - a) squared divided by 12, a standard result derived from its constant density.

Correct answer: 1

The variance is 1. Standardizing by dividing the centered variable by its standard deviation ($\sqrt{9}$, which is 3) always produces a variable with variance 1.

Correct answer: The maximum of (X - 100000) and 0

The reinsured payment equals the maximum of (X - 100000) and 0. This excess-of-loss layer pays the amount by which the loss exceeds the retention and pays nothing when the loss is below it.

Correct answer: 0.8

The expected value is 0.8. E(X) is found by integrating x times f(x) over the support: the integral of $x \cdot 4x^{3}$ from 0 to 1 equals the integral of $4x^{4}$, which is $\frac{4}{5} = 0.8$. The other values come from misapplying the density or confusing the result with the mode at x = 1.

Correct answer: 0.667

$E(X^{2})$ equals 0.667. It is computed as the integral of $x^{2} \cdot 4x^{3}$ from 0 to 1, which is the integral of $4x^{5}$, equal to $\frac{4}{6} = \frac{2}{3} = 0.667$. The value 0.64 incorrectly squares E(X), and 0.8 is E(X) itself rather than the second moment.

Correct answer: 0.2

The constant c equals 0.2. For a valid density the total area must equal 1, so c times the interval length (8 - 3 = 5) equals 1, giving c = $\frac{1}{5}$ = 0.2. The value 0.125 would correspond to a width of 8 rather than 5, and 5 is the width itself, not the density height.

Correct answer: Exponential distribution

A constant hazard rate corresponds to the exponential distribution. Its memoryless property produces a hazard function h(x) = lambda that does not depend on x, meaning the instantaneous failure rate is the same at every age. The normal, uniform, and beta distributions all have hazard rates that change with x.

Correct answer: 8

E(X) equals 8. The first moment is M'(0); differentiating gives M'(t) = 8$(1 - 4t)^{-3}$, and evaluating at t = 0 yields 8. This MGF is that of a gamma distribution with shape 2 and scale 4, whose mean alpha times theta equals 2 times 4 = 8. The value 4 confuses the scale with the mean.

Correct answer: $e^{-2}$

The probability is $e^{-2}$. For an exponential with mean 10 the rate is lambda = $\frac{1}{10}$, and the survival function P(X > x) = $e^{-x/10}$. Evaluating at x = 20 gives $e^{-\frac{20}{10}}$ = $e^{-2}$. The expression 1 - $e^{-2}$ is the CDF (probability of being at or below 20), not the survival probability.

Correct answer: 4.33

The standard deviation is 4.33. The variance of a binomial is n p (1 - p) = 100 times 0.25 times 0.75 = 18.75, and the standard deviation is $\sqrt{18.75}$, which is about 4.33. The value 18.75 is the variance, and 25 is the mean n p.

Correct answer: 33.33

The variance of Y is 33.33. The variance of a uniform on (0, 10) is $(10 - 0)^{2}$ / 12 = $\frac{100}{12}$ = 8.33. Multiplying by 2 scales the variance by $2^{2}$ = 4, and adding the constant 3 has no effect, so Var(Y) = 4 times 8.33 = 33.33. The value 8.33 is Var(X) before scaling.

Correct answer: 144

The variance of Y is 144. Variance scales by the square of the multiplicative constant, so Var(-4X + 2) = $(-4)^{2}$ times Var(X) = 16 times 9 = 144. Variance is never negative, ruling out -144, and the additive constant 2 does not change the variance.

Correct answer: 6

The variance is 6. First E(X) = 0 times 0.6 + 5 times 0.4 = 2, and E($X^{2}$) = 0 times 0.6 + 25 times 0.4 = 10. The variance is $E(X^{2}) - [E(X)]^2 = 10 - 4 = 6$. The value 2 is the mean, and 10 is the second moment, not the variance.

Correct answer: 6

The variance is 6. A defining feature of the Poisson distribution is that its mean and variance are both equal to lambda, so a mean of 6 implies a variance of 6. The value 2.45 is the standard deviation ($\sqrt{6}$), and 36 incorrectly squares the mean.

Correct answer: 0.25

The probability is 0.25. The survival function P(X > x) = 1 - F(x) = $(4/x)^{2}$, and at x = 8 this equals $(\frac{4}{8})^{2}$ = $(0.5)^{2}$ = 0.25. The value 0.75 is F(8), the probability of being at or below 8, which is the complement.

Correct answer: 4

The expected number of failures is 4. For the failures-before-first-success form of the geometric distribution, E(X) = (1 - p)/p = 0.8/0.2 = 4. The value 5 corresponds to the alternative form counting the total trials (1/p), which includes the successful trial.

Correct answer: 29

$E(X^{2})$ equals 29. Using the identity $E(X^{2}) = Var(X) + [E(X)]^2$, and noting $Var(X) = 2^{2} = 4$, gives $E(X^{2}) = 4 + 25 = 29$. The value 25 is the squared mean alone, and 21 incorrectly subtracts rather than adds the variance.

Correct answer: 0.3

The probability of no payment is 0.3. A payment of zero occurs when the loss falls at or below the deductible of 300. For a uniform loss on (0, 1000), P(X less than or equal to 300) = $\frac{300}{1000}$ = 0.3. The value 0.7 is the probability that a payment is made.

Correct answer: 1.386

The median is about 1.386. This is an exponential distribution with rate 0.5 (mean 2). The median m satisfies F(m) = 1 - $e^{-0.5m}$ = 0.5, giving $e^{-0.5m}$ = 0.5 and m = ln(2)/0.5 = 0.693/0.5 = 1.386. The value 2 is the mean, which exceeds the median for this right-skewed distribution.

Correct answer: 3

The expected number of failures is 3. For this form of the negative binomial, E(X) = r(1 - p)/p = 3 times 0.5/0.5 = 3. The value 6 would result from using the total-trials form r/p = 3/0.5, which counts the successful trials as well.

Correct answer: 0.25

The probability is 0.25. Integrating the density from 0 to 2 gives the integral of ($\frac{1}{8}$)x, which is ($\frac{1}{16}$)$x^{2}$ evaluated from 0 to 2, equal to $\frac{4}{16}$ = 0.25. The value 0.5 would result from treating the density as uniform rather than increasing linearly.

Correct answer: Poisson with mean 5

X follows a Poisson distribution with mean 5. The MGF of a Poisson with parameter lambda is $e^{\lambda(e^{t} - 1)}$, and matching gives lambda = 5. The binomial MGF has the form $(1 - p + p\, e^{t})^{n}$, and the exponential and normal MGFs do not involve $e^{t}$ inside an exponent of this form.

Correct answer: 1 - $e^{-1}$

The probability is 1 - $e^{-1}$, about 0.632. For an exponential with mean 4 the CDF is F(x) = 1 - $e^{-x/4}$, and at x = 4 (the mean) this equals 1 - $e^{-1}$. Because the exponential is right-skewed, this probability exceeds 0.5, so the median is below the mean. The expression $e^{-1}$ is the survival probability P(X > 4).

Correct answer: 81

The variance of Y is 81. First $Var(X) = E(X^{2}) - [E(X)]^2 = 13 - 4 = 9$. Scaling by 3 multiplies the variance by $3^{2} = 9$, so Var(Y) = 9 times 9 = 81. The value 9 is Var(X) before scaling, and 27 incorrectly scales by 3 instead of 9.

Correct answer: 6

The value is 6. The mean of a binomial is n p = 12 times 0.5 = 6, and for a symmetric binomial (p = 0.5) the mode coincides with the mean at the center of the distribution. The value 12 is the number of trials, and 0.5 is the success probability.

Correct answer: 500 times $e^{-0.4}$

The expected payment per loss is 500 times $e^{-0.4}$, about 335. For an exponential loss with mean theta and ordinary deductible d, the expected payment per loss equals theta times $e^{-d/\theta}$ due to the memoryless property; here theta = 500 and d/theta = $\frac{200}{500}$ = 0.4. The value 300 is simply mean minus deductible, which ignores losses falling below the deductible.

Correct answer: 0.025

The probability is approximately 0.025. For the standard normal distribution, P(Z less than 1.96) is about 0.975, so the upper-tail probability P(Z greater than 1.96) = 1 - 0.975 = 0.025. The value 0.05 is the two-tailed probability beyond plus or minus 1.96, not the single upper tail.

Correct answer: 0.0375

The variance is 0.0375. $E(X) =$ integral of $x \cdot 3x^{2} = \frac{3}{4} = 0.75$, and $E(X^{2}) =$ integral of $x^{2} \cdot 3x^{2} = \frac{3}{5} = 0.6$. The variance is $0.6 - (0.75)^{2} = 0.6 - 0.5625 = 0.0375$. The value 0.5625 is the squared mean, and 0.6 is the second moment.

Correct answer: 2

The expected value is 2. This is a geometric distribution (counting total trials to first success) with success probability p = 0.5, whose mean is 1/p = 1/0.5 = 2. The value 0.5 is the success probability itself, not the expected number of trials.

Correct answer: 500

The largest such loss is 500. The payment equals loss minus deductible until it reaches the policy limit of 400; that occurs when the ground-up loss equals deductible plus limit = 100 + 400 = 500. Beyond 500 the payment stays capped at 400. The value 400 is the maximum payment, not the loss at which capping begins.

Correct answer: 1.5

The constant c equals 1.5. The integral of (1 - $x^{2}$) from 0 to 1 is x - $x^{3}$/3 evaluated at the endpoints, which is 1 - $\frac{1}{3}$ = 2/3. Setting c times $\frac{2}{3}$ = 1 gives c = $\frac{3}{2}$ = 1.5. The value 0.667 is the integral itself, the reciprocal of the correct constant.

Correct answer: 4.5 $e^{-3}$

The probability is 4.5 $e^{-3}$. Using the Poisson mass function $e^{-\lambda}$ $\lambda^{k}$ / k! with lambda = 3 and k = 2 gives $e^{-3}$ times $\frac{9}{2}$ = 4.5 $e^{-3}$. The expression 9 $e^{-3}$ omits division by 2! = 2, and 3 $e^{-3}$ is the probability of exactly one event.