Question

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 exactly four boys in ten births. round the answer to the nearest thousandth.
a. 0.205
b. 0.040
c. 0.400
d. 13.125

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successful trials, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Determine the values of \(n\), \(k\), and \(p\)

Here, \(n = 10\) (number of births), \(k = 4\) (number of boys), and \(p=0.5\) (probability of a boy, since male and female births are equally likely)

Step3: Calculate the combination \(C(n,k)\)

$$
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$$

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

\(p^{k}=(0.5)^{4}=0.0625\), \((1 - p)^{n - k}=(0.5)^{6}=0.015625\)

Step5: Calculate the probability \(P(X = 4)\)

$$
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$$

Step6: Round to the nearest thousandth

Rounding \(0.205078125\) to the nearest thousandth gives \(0.205\)

Answer:

A. \(0.205\)