If a couple plans to have 5 children, what is the probability that there will be at least one...

Question:

If a couple plans to have 5 children, what is the probability that there will be at least one boy?

Assume boys and girls and equally likely. Is that probability high enough for the couple to be very confident that they will get at least one boy in 5 children?

The probability is can the couple be very confident that they will have at least one boy?

a.Yes because the probability is close to 0.

b.Yes because the probability is close to 1.

c.No because the probability is close to 1.

d.No because the probability is close to 0.

Binomial Probabilities

A Binomial Probability is a specific type of probability calculation that is used when an experimenter knows the probability of success of a given event, and is determining the number of successes (X) that the event occurs in a given amount of trials (n). For example, this could be used in Sports. If we know the probability that a basketball player makes a shot, we could determine the probability that this player makes a basket if he takes 10 shots. In this example, we will use binomial probability to determine the probability of a family having a boy.

To use binomial probability, we need to identify the equation that is used:

{eq}\text{binomial probability}=\binom{n}{x}\theta ^{x}(1-\theta)^{^{n-x}} {/eq}

We were given everything that we need to complete this problem. The probability of having a boy or girl is equally likely at 50%. The number of trials is 5, because the couple plans to have 5 children. X is 1, because we are determining the probability that the couple will have at least 1 boy. Since we are basically looking at the probability that the couple has more than 0 boys, it is easier to subtract the probability that the couple has 0 boys from 1. With this information we can calculate the probability as shown below:

Given:
n 5
X 1
Theta 0.5

{eq}\text{binomial probability}=\binom{n}{x}\theta ^{x}(1-\theta)^{^{n-x}}\\ \ \\ =1-\binom{5}{0}(0.5 ^{0})(1-0.5)^{^{5-0}}\\ \ \\ =1-(1)(0.5)^{^{5}}\\ \ \\ =0.969\\ \ \\ {/eq}

So, the correct answer to this question would be b, because this probability is very close to 1. A probability that is closer to 1, indicates that there is a greater probability of a success.