Question

At Least One. In Exercises 5–12, find the probability.

Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). Among the next six randomly selected births in the United States, what is the probability that at least one of them is a girl?

Short answer

The probability of having at least one girl out of six births is equal to 0.982.

Step-by-step solution

Given information

Six baby births are randomly selected.

The probability of a baby being a boy is equal to 0.512.

Probability of “at least” one

The probability of “at least one” occurrence of a given event is the probability that the event has occurred once or more than once.

Moreover, the occurrence of an event at least once complements the non-occurrence of the event. Mathematically,

$P\left(\text{occurrence}\text{of}\text{A}\text{at}\text{least}\text{once}\right)=1-P\left(\text{non - occurrence}\text{of}\text{A}\right)$

Compute the probability that at least one of them is a girl

Let A be the event of having a baby boy.

Then, is equal to 0.512.

Six random births are considered.

The probability of having no girl is the same as the event that all six are boys. This is computed as follows:

$$\begin{gathered} P\left(\text{no}\text{girl}\right)=\left(0.512\right)\times \left(0.512\right)\times \left(0.512\right)\times \left(0.512\right)\times \left(0.512\right)\times \left(0.512\right) \\ =0.0180 \end{gathered}$$

The probability of having at least one girl is equal to one minus the probability of having no girl in six births.

$$\begin{gathered} P\left(\text{at}\text{least}\text{1}\text{girl}\right)=1-P\left(\text{no}\text{girl}\right) \\ =1-0.01801 \\ =0.982 \end{gathered}$$

Therefore, the probability of having at least one girl is equal to 0.982.