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A manager records the number of hours, [tex]\( X \)[/tex], each employee works on his or her shift and develops the probability distribution below. Fifty people work for the manager. How many people work 4 hours per shift?

[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Probability Distribution} \\
\hline
Hours Worked: \( X \) & Probability: \( P(X) \) \\
\hline
3 & 0.1 \\
\hline
4 & ? \\
\hline
5 & 0.14 \\
\hline
6 & 0.3 \\
\hline
7 & 0.36 \\
\hline
8 & 0.06 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\text{Total number of employees} = 50
\][/tex]



Answer :

GinnyAnswer
To determine how many people work 4 hours per shift, we need to follow these steps:

1. Identify the total number of employees:
- The manager has a total of 50 employees.

2. List the given probabilities:
- Probability of working 3 hours: [tex]\( P(X=3) = 0.1 \)[/tex]
- Probability of working 5 hours: [tex]\( P(X=5) = 0.14 \)[/tex]
- Probability of working 6 hours: [tex]\( P(X=6) = 0.3 \)[/tex]
- Probability of working 7 hours: [tex]\( P(X=7) = 0.36 \)[/tex]
- Probability of working 8 hours: [tex]\( P(X=8) = 0.06 \)[/tex]

3. Sum up the probabilities for the given hours:
[tex]\[ P(X=3) + P(X=5) + P(X=6) + P(X=7) + P(X=8) = 0.1 + 0.14 + 0.3 + 0.36 + 0.06 = 0.96 \][/tex]

4. Calculate the missing probability for 4 hours:
- The total probability must sum up to 1 (since it represents the whole sample space).
[tex]\[ P(X=4) = 1 - \left( P(X=3) + P(X=5) + P(X=6) + P(X=7) + P(X=8) \right) = 1 - 0.96 = 0.04 \][/tex]

5. Calculate the number of employees working 4 hours:
- Multiply the probability of working 4 hours by the total number of employees:
[tex]\[ \text{Number of employees working 4 hours} = P(X=4) \times \text{Total employees} = 0.04 \times 50 = 2 \][/tex]

Therefore, the number of employees who work 4 hours per shift is [tex]\( 2 \)[/tex].

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