YouThe probability that a soccer player scores when he makes a shot is 0.527. If he takes 15 shots next game, what is the probability that he will score 3 times?
Question
YouThe probability that a soccer player scores when he makes a shot is 0.527. If he takes 15 shots next game, what is the probability that he will score 3 times?
Solution
This is a binomial probability problem. The binomial probability formula is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
In this case:
- n = 15 (the number of shots)
- k = 3 (the number of successful shots)
- p = 0.527 (the probability of scoring on a single shot)
First, calculate C(n, k). This is the number of combinations of 15 items taken 3 at a time.
C(n, k) = n! / [k!(n-k)!] = 15! / [3!(15-3)!] = 455
Next, calculate p^k, the probability of success on a single trial raised to the number of successes.
p^k = (0.527)^3 = 0.146
Then, calculate (1-p)^(n-k), the probability of failure on a single trial raised to the number of failures.
(1-p)^(n-k) = (1-0.527)^(15-3) = 0.473^12 = 0.000006
Finally, multiply these three values together to find the probability of scoring exactly 3 times in 15 shots.
P(X=3) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 455 * 0.146 * 0.000006 = 0.0004
So, the probability that the player will score exactly 3 times in 15 shots is 0.0004, or 0.04%.
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