Problem

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?

Solution

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 \(\boxed{0.78125}\).

From Solvely APP
Source: https://solvelyapp.com/problems/7548/

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