Question

The demand for facial masks among Rocky’s family varies randomly and it is independent of previous occurrences, but the average demand for the masks is 2.8 masks per week (7 days). If a new order of masks is placed on Tuesday morning, then the newly ordered masks will be delivered by CC Express on Friday morning. Rocky order 6 masks this past Tuesday morning when he found that there was one mask left.

(a) Find the probability that one mask will still be in stock on Friday morning upon the delivery of the newly ordered masks.

(b) Find the probability that Rocky’s stock of masks is inadequate to meet his family’s demand by Friday morning upon the delivery of the newly ordered masks.

(c) Find the probability that one mask will be available this Friday morning upon the delivery of the newly ordered masks and there will be at least five masks available next Tuesday morning.

Answer 1

Answer:

Given,

mean = 2.8/7 = 0.4

X ~ Poisson(0.4t)

a)

Here = 0.4*3 = 1.2

P(X = 0) = e^-1.2*1.2^0/0!

= 0.3012

b)

P(X > 1) = 1 - P(X <= 1)

= 1 - [P(0) + P(1)]

= 1 - [e^-1.2*1.2^0/0! + e^-1.2/1.2^1/1!]

= 1 - [0.3012 + 0.3614]

= 0.3374

c)

Here = 0.4*4

= 1.6

P(Y > 2) = 1 - P(Y <= 2)

= 1 - [P(0) + P(1) + P(2)]

= 1 - [e^-1.6*1.6^0/0! + e^-1.6*1.6^1/1! + e^-1.6*1.6^2/2!]

= 1 - 0.2019 - 0.3230 - 0.2584

= 0.2167

Required probability = 0.3012*0.2167

= 0.0653