Answer:
The probability of winning a game is 0.38. We want to find the probability that it takes exactly 6 games until you win.
To solve this problem, we can use the binomial probability formula. The formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes,
- n is the total number of trials,
- p is the probability of success in each trial,
- (n C k) represents the combination of n items taken k at a time.
In this case, we want to find the probability of winning on the 6th game, which means we have 5 losses before the win. The formula becomes:
P(X = 6) = (6 C 5) * (0.38)^5 * (1 - 0.38)^(6 - 5)
To calculate (6 C 5), we use the combination formula:
(6 C 5) = 6! / (5! * (6 - 5)!)
Simplifying further:
P(X = 6) = 6 * (0.38)^5 * (0.62)^1
Calculating this expression will give us the probability that it takes 6 games until you win.
2. The probability that a friend agrees to go to the movies with you is 0.27. We want to find the probability that the 5th friend you ask will be the first one to agree.
To solve this problem, we can use the geometric probability formula. The formula is:
P(X = k) = (1 - p)^(k - 1) * p
Where:
- P(X = k) is the probability that the first success occurs on the kth trial,
- p is the probability of success in each trial.
In this case, we want to find the probability that the first agreement occurs on the 5th trial. The formula becomes:
P(X = 5) = (1 - 0.27)^(5 - 1) * 0.27
Simplifying further:
P(X = 5) = 0.73^4 * 0.27
Calculating this expression will give us the probability that the 5th friend you ask will be the first one to agree.
Please let me know if there is anything else I can help you with.
Explanation: