Four letters to different insureds are prepared along with accompanying envelopes. The letters are put into the envelopes randomly.

Calculate the probability that at least one letter ends up in its accompanying envelope.

• 27/256
• 1/4
• 11/24
• 5/8
• 3/4

A drawer contains four pairs of socks, with each pair a different color. One sock at a time is randomly drawn from the drawer until a matching pair is obtained. Calculate the probability that the maximum number of draws is required.

• 0.0006
• 0.0095
• 0.0417
• 0.1429
• 0.2286

On a block of ten houses, [math]k[/math] are not insured. A tornado randomly damages three houses on the block. The probability that none of the damaged houses are insured is 1/120.

Calculate the probability that at most one of the damaged houses is insured.

• 1/5
• 7/40
• 11/60
• 49/60
• 119/120

Four distinct integers are chosen randomly and without replacement from the first twelve positive integers. Let [math]X[/math] be the random variable representing the second largest of the four selected integers, and let [math]p[/math] be the probability function for [math]X[/math].

Determine [math]p(x)[/math], for integer values of [math]x[/math], where [math]p(x)\gt0[/math].

• [math]\frac{( x − 1)( x − 2)(12 − x)}{990}[/math]
• [math]\frac{( x − 1)( x − 2)(12 − x)}{495}[/math]
• [math]\frac{( x − 1)(12 − x)(11 − x)}{495}[/math]
• [math]\frac{( x − 1)(12 − x)(11 − x)}{990}[/math]
• [math]\frac{(10 − x)(12 − x)(11 − x)}{990}[/math]

A state is starting a lottery game. To enter this lottery, a player uses a machine that randomly selects six distinct numbers from among the first 30 positive integers. The lottery randomly selects six distinct numbers from the same 30 positive integers. A winning entry must match the same set of six numbers that the lottery selected. The entry fee is 1, each winning entry receives a prize amount of 500,000, and all other entries receive no prize.

Calculate the probability that the state will lose money, given that 800,000 entries are purchased.

• 0.33
• 0.39
• 0.61
• 0.67
• 0.74



Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

From 27 pieces of luggage, an airline luggage handler damages a random sample of four. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of the damaged pieces are insured.

Calculate the probability that exactly two of the four damaged pieces are insured.

• 0.06
• 0.13
• 0.27
• 0.30
• 0.31

Thirty items are arranged in a 6-by-5 array as shown.

Calculate the number of ways to form a set of three distinct items such that no two of the selected items are in the same row or same column.

• 200
• 760
• 1200
• 4560
• 7200

In a certain game of chance, a square board with area 1 is colored with sectors of either red or blue. A player, who cannot see the board, must specify a point on the board by giving an x-coordinate and a y-coordinate. The player wins the game if the specified point is in a blue sector. The game can be arranged with any number of red sectors, and the red sectors are designed so that

where [math]R_i[/math] is the area of the [math]i^{\textrm{th}}[/math] red sector.

• 3
• 4
• 5
• 6
• 7

A deck of ordinary cards is shuffled and 13 cards are dealt. What is the probability that the last card dealt is an ace?

• 0.057
• 0.077
• 0.097
• 0.11
• 0.12

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.