Final answer:
The probability questions on roulette betting can be solved using the binomial distribution formula. For exactly 8 bets to achieve 5 wins, the associated probability involves a combination of wins and losses in those bets. For at least 9 bets, the probability is obtained by subtracting the cumulative probability of stopping within 8 bets from 1.
Step-by-step explanation:
Probability of Winning Roulette Bets
To answer these questions, we need to use concepts from probability, particularly the binomial probability formula which is given by P(X = k) = (n choose k) · p^k · (1-p)^(n-k), where n is the number of trials, k is the number of desired success, p is the probability of success on a single trial, and (1-p) is the probability of failure.
a) The probability that the gambler needs to make exactly 8 bets to win 5 times can be found by considering that the gambler wins 4 times and loses 3 times in the first 7 bets and wins the 8th bet. Using the formula:
P(X = 5) = (7 choose 4) · (18/37)^4 · (19/37)^3 · (18/37)
b) To calculate the probability of requiring at least 9 bets, we subtract the probability of stopping within 8 bets from 1:
P(X ≥ 9) = 1 - P(X ≤ 8)
where P(X ≤ 8) is the sum of probabilities of stopping at each of the number of 5 wins from 5 to 8 bets.