through (d) below.
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a. Find the probability of getting exactly 6 girls in 8 births.
0.117 (Type an integer or a decimal. Do not round.)
b. Find the probability of getting 6 or more girls in 8 births.
(Type an integer or a decimal. Do not round.)
Answer
Final Answer: The probability of getting exactly 6 girls in 8 births is \(\boxed{0.109375}\).
Step 1 ：The problem is asking for the probability of getting exactly 6 girls in 8 births. This is a binomial probability problem, where the number of trials is 8 (the number of births), the number of successful outcomes we're interested in is 6 (the number of girls), and the probability of success on each trial is 0.5 (since the probability of having a girl or a boy is assumed to be equal).
Step 2 ：The formula for binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability we're trying to find, \(C(n, k)\) is the number of combinations of n items taken k at a time, p is the probability of success on each trial, n is the number of trials, and k is the number of successful outcomes we're interested in.
Step 3 ：In this case, n=8, k=6, and p=0.5.
Step 4 ：Calculating the combinations, \(C(n, k)\), we get 28.
Step 5 ：Calculating \(p^k\), we get 0.015625.
Step 6 ：Calculating \((1-p)^(n-k)\), we get 0.25.
Step 7 ：Multiplying these values together, we get the probability of getting exactly 6 girls in 8 births, which is 0.109375.
Step 8 ：The calculated probability is 0.109375, which is not equal to the given answer of 0.117. This discrepancy could be due to rounding errors in the given answer, or it could be that the probability of having a girl is not exactly 0.5. However, since the question does not provide any other information, we will assume that the calculated probability is the correct answer.
Step 9 ：Final Answer: The probability of getting exactly 6 girls in 8 births is \(\boxed{0.109375}\).
