Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three boys in ten births.

Question

Quizplus · Accepted Answer

We can use the binomial distribution to solve this problem. Let X be the number of boys in ten births. Then X follows a binomial distribution with parameters n=10 and p=0.5. The probability of at most three boys in ten births is:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Using the binomial probability formula, we get:

P(X = k) = (10 choose k) * 0.5^k * 0.5^(10-k)

Thus,

P(X = 0) = (10 choose 0) * 0.5^0 * 0.5^10 = 0.00098
P(X = 1) = (10 choose 1) * 0.5^1 * 0.5^9 = 0.00977
P(X = 2) = (10 choose 2) * 0.5^2 * 0.5^8 = 0.04395
P(X = 3) = (10 choose 3) * 0.5^3 * 0.5^7 = 0.11719

Therefore,

P(X ≤ 3) = 0.00098 + 0.00977 + 0.04395 + 0.11719 = 0.172

So the answer is A, 0.172. The number of ways to have at most three boys in ten births is the sum of the number of ways to have 0, 1, 2, or 3 boys in ten births.

The number of ways to have 0 boys in ten births is $\binom{10}{0} = 1$. The number of ways to have 1 boy in ten births is $\binom{10}{1} = 10$. The number of ways to have 2 boys in ten births is $\binom{10}{2} = 45$. The number of ways to have 3 boys in ten births is $\binom{10}{3} = 120$.

The total number of ways to have any combination of boys and girls in ten births is $2^{10} = 1024$.

So the probability of at most three boys in ten births is $$\frac{1 + 10 + 45 + 120}{1024} = \frac{196}{1024} \approx \boxed{0.172}$$

Assume That Male and Female Births Are Equally Likely and That