Problem Slot machines have three drums with the same number of positions on each

Question

Problem Slot machines have three drums with the same number of positions on each drum. Each position contains a symbol. When you pull the lever on the machine, the drums spin. Assume that each drum is equally likely to stop in any position, and that the ositions in which the three drums stop are independent of each other and independent from spin to spin. Suppose a particular slot machine has 8 positions on each drum, and that each position contains one of two symbols: an eggplant or a grapefruit. You win the jackpot if all three drums stop with eggplants showing. The first drum has 5 eggplants and 3 grapefruits, the second has 7 eggplants and 1 grapefruit, and the third drum has 3 eggplants and 5 grapefruits. In each pull of the lever, the probability of winning the jackpot is The number of times I win the jackpot in 15 attempts has (02) The probablity of winning the jackpot at least once in 15 attempts is (Q3) The number of attempts it takes to win the first jackpot has (Q4) The probability that it takes exactly 15 attempts to win the jackpot the first time is (Q5) The number of attempts it takes to win the jackpot for the seventh time has (Q6) The chance that I win the jackpot for the seventh time on my fifteenth attempt is (Q7)

Explanation / Answer

(Q1) Prob. = (5/8) * (7/8) * (3/8) = 0.2051.

(Q2) Number of times = 15 * 0.2051 = 3 (approx.)

(Q3) Prob. = 0.968

(Q4) Number of attempts = 1/0.2051 = 4.87 = 4 (approx.) or 5 (approx.)

(Q5) Prob. = 0.0083

(Q6) Number of attempts = 2.063 = 2 (approx.)

(Q7) Prob. = 0.0083

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