To determine the probability that a teenager has exactly 3 pairs of shoes in their closet, follow these steps:
1. Identify the frequencies of the pairs of shoes:
- 1 pair: 18 teenagers
- 2 pairs: 30 teenagers
- 3 pairs: 57 teenagers
- 4 pairs: 30 teenagers
- 5 pairs: 15 teenagers
2. Calculate the total number of teenagers surveyed:
Add up all the frequencies:
[tex]\[
\text{Total number of teenagers} = 18 + 30 + 57 + 30 + 15 = 150
\][/tex]
3. Identify the frequency of teenagers having exactly 3 pairs of shoes:
From the given data, the frequency is:
[tex]\[
\text{Frequency of 3 pairs of shoes} = 57
\][/tex]
4. Calculate the probability that a teenager has exactly 3 pairs of shoes:
The probability [tex]\( P(3) \)[/tex] is obtained by dividing the frequency of teenagers with exactly 3 pairs of shoes by the total number of teenagers surveyed:
[tex]\[
P(3) = \frac{\text{Frequency of 3 pairs of shoes}}{\text{Total number of teenagers}}
\][/tex]
[tex]\[
P(3) = \frac{57}{150}
\][/tex]
5. Simplify the probability (if necessary):
In this case, simplifying the fraction gives:
[tex]\[
P(3) = \frac{57}{150} \approx 0.38
\][/tex]
Therefore, the probability that a teenager has exactly 3 pairs of shoes is:
[tex]\[
P(3) = 0.38
\][/tex]
