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].
