Question

a group of 126 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.

	doctor	nurse
afternoon	26	18
evening	7	23

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 doctor or worked the evening shift?
(b) what is the probability that the volunteer worked the morning shift, given that the volunteer is a nurse?

Explanation:

Step1: Calculate total number of volunteers

The total number of volunteers is \(23 + 29+18 + 26+23 + 7=126\).

Step2: Calculate number of doctors or evening - shift workers for part (a)

Number of doctors is \(29 + 26+7 = 62\), number of evening - shift workers is \(23 + 7=30\), and number of doctor - evening - shift workers is \(7\). Using the inclusion - exclusion principle, the number of doctors or evening - shift workers is \(62+30 - 7=85\). The probability that the volunteer is a doctor or worked the evening shift is \(\frac{85}{126}\approx0.67\).

Step3: Calculate number of nurses and morning - shift workers for part (b)

Number of nurses is \(23 + 18+23 = 64\), number of morning - shift workers is \(23 + 29 = 52\), and number of nurse - morning - shift workers is \(23\). The number of nurses and morning - shift workers is \(23\). The probability that the volunteer is a nurse and worked the morning shift given that the volunteer is a nurse: First, the probability of being a nurse is \(\frac{64}{126}\), and the probability of being a nurse and working the morning shift is \(\frac{23}{126}\). By the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), here \(A\) is working the morning shift and \(B\) is being a nurse. So \(P=\frac{23}{64}\approx0.36\).

Answer:

(a) \(0.67\)
(b) \(0.36\)