To determine the probability that a teenager has exactly 4 pairs of shoes, we'll follow these steps:
1. Identify the total number of teenagers surveyed.
2. Find the number of teenagers who have exactly 4 pairs of shoes.
3. Calculate the probability using the ratio of teenagers with exactly 4 pairs of shoes to the total number of teenagers.
### Step 1: Total Number of Teenagers
First, sum up the frequencies in the provided table to find the total number of teenagers:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Pairs of Shoes} & \text{Frequency} \\
\hline
1 & 18 \\
2 & 30 \\
3 & 57 \\
4 & 30 \\
5 & 15 \\
\hline
\end{array}
\][/tex]
[tex]\[
\text{Total number of teenagers} = 18 + 30 + 57 + 30 + 15 = 150
\][/tex]
### Step 2: Frequency of Teenagers with Exactly 4 Pairs of Shoes
From the table, the number of teenagers with exactly 4 pairs of shoes is:
[tex]\[
\text{Frequency of 4 pairs of shoes} = 30
\][/tex]
### Step 3: Calculate the Probability
The probability [tex]\( P(4) \)[/tex] that a teenager has exactly 4 pairs of shoes is calculated by dividing the frequency of teenagers with 4 pairs of shoes by the total number of teenagers:
[tex]\[
P(4) = \frac{\text{Frequency of 4 pairs of shoes}}{\text{Total number of teenagers}} = \frac{30}{150}
\][/tex]
Simplifying this fraction:
[tex]\[
P(4) = \frac{30}{150} = \frac{1}{5} = 0.2
\][/tex]
### Conclusion
The probability that a teenager has exactly 4 pairs of shoes in their closet is:
[tex]\[
P(4) = 0.2
\][/tex]
