A roulette wheel has alternating red and black

numbered slots into one of which the ball finally
stops to determine the winner. If a gambler always
bets on black to win, what is the probability of
winning at least 24 times in a series of 36 spins?
(Note that at least 24 wins means 24 or more.)

a roulette wheel has alternating red and black numbered slots in one of which the ball finally stops to determine the winner. It gambler always bets on black to win. What is the probability at least 24 times in a series of 36?

To find the probability of winning at least 24 times in a series of 36 spins when betting on black, we can use the binomial probability formula. This formula calculates the probability of getting a certain number of successes in a fixed number of independent Bernoulli trials.

In this case, each spin of the roulette wheel is an independent trial with two possible outcomes (winning on black or not winning on black). The probability of winning on black in a single spin is 18/38 (since there are 18 black slots out of a total of 38 slots).

Now, we need to calculate the probability of getting 24, 25, 26, ..., up to 36 successes in 36 spins. Since we want the probability of winning at least 24 times, we sum up the probabilities for 24, 25, ..., 36.

P(X ≥ 24) = P(X = 24) + P(X = 25) + ... + P(X = 36)

P(X = k) = C(n, k) * p^k * q^(n-k)

Where:
- n is the total number of trials (36 spins)
- k is the number of successes (from 24 to 36 wins)
- p is the probability of winning on black (18/38)
- q is the probability of not winning on black (20/38)

Using these values in the formula, we can calculate the probabilities of each individual outcome and sum them up to get the final result.

P(X ≥ 24) = P(X = 24) + P(X = 25) + ... + P(X = 36)