Question

rachel found out the number of members in all the clubs at her school. number of members 3 3 3 12 12 12 12 36 36 36. x is the number of members that a randomly chosen club has. what is the expected value of x? write your answer as a decimal.

Explanation:

Step1: Calculate total number of clubs

There are 3 clubs with 3 members, 4 clubs with 12 members and 3 clubs with 36 members. So the total number of clubs is $3 + 4+3=10$.

Step2: Calculate probability for each number of members

The probability of a club having 3 members, $P(3)=\frac{3}{10}=0.3$. The probability of a club having 12 members, $P(12)=\frac{4}{10} = 0.4$. The probability of a club having 36 members, $P(36)=\frac{3}{10}=0.3$.

Step3: Calculate expected - value formula

The formula for the expected value $E(X)=\sum_{i}x_{i}P(x_{i})$. Here $x_1 = 3$, $P(x_1)=0.3$, $x_2 = 12$, $P(x_2)=0.4$, $x_3 = 36$, $P(x_3)=0.3$.

Step4: Calculate the expected value

$E(X)=3\times0.3 + 12\times0.4+36\times0.3=0.9 + 4.8+10.8 = 16.5$.

Answer:

$16.5$