The Smith family was one of the first to come to the U.S. They had 5 children. Assuming that the probability of a child being a girl is .5 , find the probability that the Smith family had:
at least 2 girls?
at most 4 girls?
Answer
Final Answer: The probability that the Smith family had at least 2 girls and at most 4 girls is \(\boxed{0.78125}\).
Step 1 ：The problem is asking for the probability of having at least 2 girls and at most 4 girls in a family of 5 children. This is a binomial probability problem, where the number of trials is 5 (the number of children), the probability of success (having a girl) is 0.5, and we want to find the probability of having 2 to 4 successes.
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 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, and k is the number of successes.
Step 3 ：We need to calculate this probability for k=2, k=3, k=4 and then sum these probabilities to get the final answer.
Step 4 ：Let's calculate the probabilities: for k=2, the probability is 0.3125; for k=3, the probability is 0.3125; for k=4, the probability is 0.15625.
Step 5 ：Adding these probabilities together, we get a final probability of 0.78125.
Step 6 ：Final Answer: The probability that the Smith family had at least 2 girls and at most 4 girls is \(\boxed{0.78125}\).
