Question

a group of 117 doctors and nurses volunteered to run a health fair. each volunteer worked one shift. the table below summarizes the data on the volunteers and their shifts.

	morning	afternoon	evening
nurse	12	22	9

suppose a volunteer from the health fair is chosen at random. answer each part. do not round intermediate computations, and round your answers to the nearest hundredth.

(a) what is the probability that the volunteer is a nurse or worked the morning shift?

(b) what is the probability that the volunteer is a doctor, given that the volunteer worked the afternoon shift?

Explanation:

Step1: Calculate total number of volunteers

The total number of volunteers is \(22 + 15+37+12 + 22+9=117\).

Step2: Calculate number of nurses

The number of nurses is \(12 + 22+9 = 43\).

Step3: Calculate number of morning - shift workers

The number of morning - shift workers is \(22 + 12=34\).

Step4: Calculate number of nurses who worked morning shift

The number of nurses who worked morning shift is \(12\).

Step5: Calculate probability for part (a)

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), the probability that the volunteer is a nurse or worked the morning shift is \(\frac{43 + 34-12}{117}=\frac{65}{117}\approx0.56\).

Step6: Calculate number of afternoon - shift workers

The number of afternoon - shift workers is \(15 + 22=37\).

Step7: Calculate number of doctors who worked afternoon shift

The number of doctors who worked afternoon shift is \(15\).

Step8: Calculate probability for part (b)

Using the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), the probability that the volunteer is a doctor given that the volunteer worked the afternoon shift is \(\frac{15}{37}\approx0.41\).

Answer:

(a) \(0.56\)
(b) \(0.41\)