In a casino game, a gambler selects four different numbers from the first twelve positive integers. The casino then randomly draws nine numbers without replacement from the first twelve positive integers. The gambler wins the jackpot if the casino draws all four of the gambler’s selected numbers.

Calculate the probability that the gambler wins the jackpot.

• 0.002
• 0.255
• 0.296
• 0.573
• 0.625

There are 10 applicants for the director of computing. The applicants are interviewed independently by each member of the three-person search committee and ranked from 1 to 10. A candidate will be hired if he or she is ranked first by at least two of the three interviewers. Find the probability that the search will be conclusive (a candidate is chosen) if the members of the committee really have no ability at all to judge the candidates and just rank the candidates randomly.

• 0.18
• 0.22
• 0.25
• 0.28
• 0.3

References

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

A computing center has 3 processors that receive 10 jobs, with the jobs assigned to the processors purely at random so that all of the 3¹⁰ possible assignments are equally likely. Find the probability that exactly one processor has no jobs.

1. 0.02
2. 0.03
3. 0.04
4. 0.05
5. 0.06

References

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

Charles claims that he can distinguish between beer and ale 75 percent of the time. Ruth bets that he cannot and, in fact, just guesses. To settle this, a bet is made: Charles is to be given ten small glasses, each having been filled with beer or ale, chosen by tossing a fair coin. He wins the bet if he gets seven or more correct. Find the probability that Charles wins if he has the ability that he claims.

• 0.68
• 0.7
• 0.73
• 0.75
• 0.78

References

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

Red and blue balls are placed in two urns. One urn contains 5 red balls and 8 blue balls and the other urn contains 6 red balls and 4 blue balls. Two balls are randomly selected from each urn.

Determine the probability that two balls of differing colors are observed.

• 0.87
• 0.89
• 0.91
• 0.93
• 0.95
• 0.97

An insurance company employs 10 claims adjusters at a local office where 4 adjusters deal with home insurance claims, 2 adjusters deal with auto insurance claims, and 4 adjusters deal with travel insurance claims. The local office receives 6 new reported claims from policyholders: 2 claims for auto, 3 claims for home, and 1 claim for travel.

If each reported claim is allocated to an adjuster in a random manner, determine the probability that all reported claims will be serviced by an adjuster that is qualified to service the claim.

• 0.05
• 0.0545
• 0.0622
• 0.0682
• 0.0762

An urn contains four fair dice. Two have faces numbered 1, 2, 3, 4, 5, and 6; one has faces numbered 2, 2, 4, 4, 6, and 6; and one has all six faces numbered 6. One of the dice is randomly selected from the urn and rolled. The same die is rolled a second time. Calculate the probability that a 6 is rolled both times.

• 0.174
• 0.250
• 0.292
• 0.380
• 0.417

George and Paul play a betting game. Each chooses an integer from 1 to 20 (inclusive) at random. If the two numbers differ by more than 3, George wins the bet. Otherwise, Paul wins the bet. Calculate the probability that Paul wins the bet.

• 0.27
• 0.32
• 0.40
• 0.48
• 0.66

Each week, a subcommittee of four individuals is formed from among the members of a committee comprising seven individuals. Two subcommittee members are then assigned to lead the subcommittee, one as chair and the other as secretary.

Calculate the maximum number of consecutive weeks that can elapse without having the subcommittee contain four individuals who have previously served together with the same subcommittee chair.

• 70
• 140
• 210
• 420
• 840

Bowl I contains eight red balls and six blue balls. Bowl II is empty. Four balls are selected at random, without replacement, and transferred from bowl I to bowl II. One ball is then selected at random from bowl II.

Calculate the conditional probability that two red balls and two blue balls were transferred from bowl I to bowl II, given that the ball selected from bowl II is blue.

• 0.21
• 0.24
• 0.43
• 0.49
• 0.57