Question

Every day I receive a random number of packages in the mail. I sanitize each one with hand sanitizer. I have enough hand sanitizer to sanitize 600 packages. Each day, I get at least two packages, but not more than a twelve, and I always receive an even number of packages. Among the possible numbers of packages I might receive, all are equally likely. Moreover, different days are independent. What is the approximate probability that my stockpile of hand sanitizer will last at least three months? Assume there are 30 days in every month.

Answer 1

Answer:

Given,

Let us consideer X be the No. of envelopes received

P(X = 2) = P(X = 4) = P(X = 6) = P(X = 8) = P(X = 10) = P(X = 12) = 2/6 = 1/3

Expected value E(X) = x*P(x)

= 1/6[2+4+6+8+10+12]

= 42/6

= 7

E(X^2) = x^2*P(x)

= 1/6(2^2 + 4^2 + 6^2 + 8^2 + 10^2 + 12^2)

= 60.67

Standard deviation = sqrt(60.67 - 7^2)

= 3.42

X ~ (7 , 3.42/sqrt(90)) = (7 , 0.36)

Required probability = P(X <= 600)

= P(z <= -0.926)

= 0.177223 [since from z table]

Required probability = 0.1772