Question

In Exercises 15–20, refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children.

Using Probabilities for Significant Events

a. Find the probability of getting exactly 6 girls in 8 births.

b. Find the probability of getting 6 or more girls in 8 births.

c. Which probability is relevant for determining whether 6 is a significantly high number of girls in 8 births: the result from part (a) or part (b)?

d. Is 6 a significantly high number of girls in 8 births? Why or why not?

Number of girls x	P(x)
0	0.004
1	0.031
2	0.109
3	0.219
4	0.273
5	0.219
6	0.109
7	0.031
8	0.004

Short answer

a.The probability of getting exactly 6 girls in 8 births is 0.109.

b. The probability of getting 6 or more girls in 8 births is 0.144.

c. The probability used in (b) part.

d. No, 6 is not a significantly high number of girls in 8 births.

Step-by-step solution

Given information

The probability distribution for the number of girls among 8 children is provided.

The variable x is the number of girls among 8 children.

Calculate the probability of getting exactly 6 girls in 8 births

a.

Using the probability distribution table, the probability corresponding to 6 girls in 8 births is 0.109.

Thus, the probability of getting exactly 6 girls in 8 births is 0.109.

Calculate the probability of getting 6 or more girls in 8 births

b.

Using the probability distribution table, the probability corresponding to 6 girls in 8 births is 0.109.

The probability corresponding to 7 girls in 8 births is 0.031.

The probability corresponding to 8 girls in 8 births is 0.004.

The probability of getting 7 or more girls in 8 births is computed as:

$$\begin{gathered} P\left(x⩾6\right)=P\left(x=6\right)+P\left(x=7\right)+P\left(x=8\right) \\ =0.109+0.031+0.004 \\ =0.144 \end{gathered}$$

Thus, the probability of getting 6 or more girls in 8 births is 0.144.

State the probability relevant to determine whether 6 is high number of girls in 8 births

c.

The following probability expression is used to determine if the given sample value is significantly high or not:

$P\left(x\text{or}\text{more}\right)⩽0.05$

If the above expression holds true, then the number of successes for that event can be considered significantly high.

Here, part (b) computed the probability of 6 or more girls in 8 births.

The probability computed in (b) part is relevant to determine whether 6 is a high number of girls in 8 births.

Check whether 6 is a significantly high number of girls in 8 births

d.

Since the probability of 6 or more high number of girls in 8 births is 0.144, which is not less than 0.05, thus, the number of girls equal to 6 cannot be considered significantly high.