Let's solve each part of the question step-by-step.
### Part (a)
We need to determine the probability that a randomly selected volunteer is a doctor, given that the volunteer worked the morning shift.
Step 1: Identify the number of doctors and nurses for each shift from the table:
- Morning Doctors: 36
- Morning Nurses: 11
Step 2: Calculate the total number of volunteers who worked the morning shift:
[tex]\[ \text{Total Morning Volunteers} = \text{Morning Doctors} + \text{Morning Nurses} = 36 + 11 = 47 \][/tex]
Step 3: The conditional probability [tex]\(P(\text{Doctor}|\text{Morning})\)[/tex] is calculated as the number of doctors in the morning divided by the total number of volunteers in the morning shift:
[tex]\[ P(\text{Doctor}|\text{Morning}) = \frac{\text{Morning Doctors}}{\text{Total Morning Volunteers}} = \frac{36}{47} \][/tex]
Step 4: Compute the probability and round to the nearest hundredth:
[tex]\[ P(\text{Doctor}|\text{Morning}) \approx 0.77 \][/tex]
Thus, the probability that the volunteer is a doctor given that the volunteer worked the morning shift is approximately [tex]\(0.77\)[/tex].
### Part (b)
We need to determine the probability that the volunteer is a nurse or worked the afternoon shift.
Step 1: Identify the total number of each type and shift from the table:
- Total Volunteers: 163
- Total Nurses: 11 (morning) + 42 (afternoon) + 8 (evening) = 61
- Total Afternoon Volunteers: 59 (doctors) + 42 (nurses) = 101
- Afternoon Nurses: 42
Step 2: Compute the probability of being a nurse:
[tex]\[ P(\text{Nurse}) = \frac{\text{Total Nurses}}{\text{Total Volunteers}} = \frac{61}{163} \][/tex]
Step 3: Compute the probability of working the afternoon shift:
[tex]\[ P(\text{Afternoon}) = \frac{\text{Total Afternoon Volunteers}}{\text{Total Volunteers}} = \frac{101}{163} \][/tex]
Step 4: Compute the probability of being a nurse and working the afternoon shift (which is the event intersection):
[tex]\[ P(\text{Nurse and Afternoon}) = \frac{\text{Afternoon Nurses}}{\text{Total Volunteers}} = \frac{42}{163} \][/tex]
Step 5: Use the formula for the union of two probabilities:
[tex]\[ P(\text{Nurse or Afternoon}) = P(\text{Nurse}) + P(\text{Afternoon}) - P(\text{Nurse and Afternoon}) \][/tex]
[tex]\[ P(\text{Nurse or Afternoon}) = \frac{61}{163} + \frac{101}{163} - \frac{42}{163} \][/tex]
Step 6: Calculate the final probability and round to the nearest hundredth:
[tex]\[ P(\text{Nurse or Afternoon}) \approx 0.74 \][/tex]
Thus, the probability that the volunteer is a nurse or worked the afternoon shift is approximately [tex]\(0.74\)[/tex].
