Question

When a customer enters a pharmacy, the probabilities that he or she will have 0, 1, or 2 or more prescrip- tions filled are 0.60, 0.25 and 0.15, respectively. For a sample of six people who enter the pharmacy, find the probability that

(a) three will have 0 prescriptions, two will have 1 prescription, and one will have 2 or more prescriptions.

(b) four will have 0 prescriptions, one will have 1 prescription, and one will have 2 or more prescriptions.

Answer 1

Multinominal Distribution:(when only three outcomes are considered)

Probability of obtaining a specific set of outcomes when there are three possible outcomes for each event:

where

p is the probability,
n is the total number of events
n₁ is the number of times Outcome 1 occurs,
n₂ is the number of times Outcome 2 occurs,
n₃ is the number of times Outcome 3 occurs,
p₁ is the probability of Outcome 1
p₂ is the probability of Outcome 2, and
p₃ is the probability of Outcome 3.

Probability that he or she will have '0' prescriptions filled : p₁ = 0.60

Probability that he or she will have '1' prescriptions filled : p₂ = 0.25

Probability that he or she will have '2 or more' prescriptions filled : p₃ = 0.15

For a sample of six people;

i.e n= 6

(a) Probability that three will have 0 prescriptions, two will have 1 prescription, and one will have 2 or more prescriptions.

n₁ : Number of people having '0' prescriptions = 3

n₂ : Number of people having '1' prescriptions = 2

n₃ : Number of people having '2 or more ' prescriptions = 1

Probability that three will have 0 prescriptions, two will have 1 prescription, and one will have 2 or more prescriptions =

Probability that three will have 0 prescriptions, two will have 1 prescription, and one will have 2 or more prescriptions = 0.1215

(b) Probability that four will have 0 prescriptions, one will have 1 prescription, and one will have 2 or more prescriptions.

n₁ : Number of people having '0' prescriptions = 4

n₂ : Number of people having '1' prescriptions = 1

n₃ : Number of people having '2 or more ' prescriptions = 1

Probability that four will have 0 prescriptions, one will have 1 prescription, and one will have 2 or more prescriptions = 0.1458