To determine how many people work 4 hours per shift, let's first find the probability of an employee working exactly 4 hours (denoted [tex]\( P(X = 4) \)[/tex]) given the provided probability distribution.
Here's the given distribution:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Hours Worked} & \text{Probability} \\
\hline 3 & 0.1 \\
\hline 4 & ? \\
\hline 5 & 0.14 \\
\hline 6 & 0.3 \\
\hline 7 & 0.36 \\
\hline 8 & 0.06 \\
\hline
\end{array}
\][/tex]
We know that the sum of probabilities for all possible values of [tex]\( X \)[/tex] must equal 1. Therefore, we can write the following equation:
[tex]\[
P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 1
\][/tex]
Plugging in the given probabilities:
[tex]\[
0.1 + P(X = 4) + 0.14 + 0.3 + 0.36 + 0.06 = 1
\][/tex]
Summing the known probabilities:
[tex]\[
0.1 + 0.14 + 0.3 + 0.36 + 0.06 = 0.96
\][/tex]
We can now solve for [tex]\( P(X = 4) \)[/tex]:
[tex]\[
P(X = 4) = 1 - 0.96 = 0.04
\][/tex]
This means that the probability of an employee working exactly 4 hours is 0.04.
Now, we need to find out how many out of the 50 employees work 4 hours per shift. We can do this by multiplying the total number of employees by the probability of working 4 hours:
[tex]\[
\text{Number of people working 4 hours} = 50 \times 0.04 = 2
\][/tex]
Therefore, the number of people who work exactly 4 hours per shift is 2.
