Let’s solve the problem step-by-step to find the probability that a teenager has exactly 3 pairs of shoes in their closet.
### Step 1: Understand the Given Data
We are provided with the distribution of pairs of shoes for teenagers. The table is as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Pairs of Shoes} & 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Frequency} & 18 & 30 & 57 & 30 & 15 \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate the Total Number of Teenagers
The total number of teenagers is the sum of the frequencies for each possible number of pairs of shoes:
[tex]\[
\text{Total Frequency} = 18 + 30 + 57 + 30 + 15
\][/tex]
From the given data, the sum is:
[tex]\[
18 + 30 + 57 + 30 + 15 = 150
\][/tex]
Thus, the total number of teenagers surveyed is 150.
### Step 3: Identify the Frequency of Teenagers with Exactly 3 Pairs of Shoes
From the table, we can see that the frequency of teenagers who have exactly 3 pairs of shoes is 57.
### Step 4: Calculate the Probability
The probability [tex]\( P(3) \)[/tex] that a teenager has exactly 3 pairs of shoes is the ratio of the number of teenagers with 3 pairs of shoes to the total number of teenagers. Mathematically, this probability is given by:
[tex]\[
P(3) = \frac{\text{Frequency of 3 pairs of shoes}}{\text{Total Frequency}}
\][/tex]
Plugging in the numbers:
[tex]\[
P(3) = \frac{57}{150}
\][/tex]
### Step 5: Simplify the Fraction
To convert this fraction into a decimal:
[tex]\[
P(3) = \frac{57}{150} \approx 0.38
\][/tex]
### Final Answer
The probability that a teenager has exactly 3 pairs of shoes in their closet is:
[tex]\[
P(3) = 0.38
\][/tex]
