You have recently got married and you plan to have 4 children. If the probability that a child is a boy is 0.5, what is the probability that you will have exactly 3 boys?
Question
You have recently got married and you plan to have 4 children. If the probability that a child is a boy is 0.5, what is the probability that you will have exactly 3 boys?
Solution
To solve this problem, we will use the binomial probability formula, which 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
- n is the number of trials
- k is the number of successes
In this case, we want to find the probability of having exactly 3 boys (k=3) out of 4 children (n=4), given that the probability of having a boy (p) is 0.5.
Step 1: Calculate C(n, k) C(4, 3) = 4! / [(4-3)! * 3!] = 4
Step 2: Calculate p^k (0.5)^3 = 0.125
Step 3: Calculate (1-p)^(n-k) (1-0.5)^(4-3) = 0.5
Step 4: Multiply the results together P(X=3) = C(4, 3) * (0.5)^3 * (0.5)^(4-3) = 4 * 0.125 * 0.5 = 0.25
So, the probability that you will have exactly 3 boys is 0.25 or 25%.
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