Question

The probability that a customer's order is not shipped on time is 0.03. A particular customer places three orders, and the orders are placed far enough apart in time that they can be considered to be independent events.

a. What is the probability that all are shipped on time?
b. What is the probability that exactly one is not shipped on time?
c. What is the probability that two or more orders are not shipped on time?

Answer 1

a. The probability is 0.912673 or 91.2673%

b. The probability is 0.084681 or 8.4681%

c. The probability is 0.002646 or 0.2646%

Explanation:

This situation can be describe with a Binomial distribution, in which we have n identical and independents events with two possible results, success and fail, with probability p and 1-p respectively. The probability of x, is given by:

`P(x;n;p)=nCx*p^(x) *(1-p)^(n-x)`

Where x is the number of events with success from the n events and nCx is calculated as:

`nCx=(n!)/(k!(n-k)!)`

In this case, n is equal to 3 because we have three orders and p is equal to 0.03 because we are going to call success when the order is not shipped on time.

Then, the probability that all are shipped on time is the probability when x is equal to zero. it is when there aren't orders not shipped on time. So, the probability is calculated as:

`P(0;3;0.03)=3C0*0.03^(0) *(1-0.03)^(3-0)`

`P(0;3;0.03)=1*1*0.912673 = 0.912673`

At the same way, we obtain the probability that exactly one is not shipped on time when we replace x by 1. That is calculated as:

`P(1;3;0.03)=3C1*0.03^(1) *(1-0.03)^(3-1)`

`P(1;3;0.03)=3*0.03*0.9409 = 0.084681`

Finally, the probability that two or more orders are not shipped on time is the probability when x is equal to 2 plus the probability when x is equal to 3.

The probability when we replace x by 2 is:

`P(2;3;0.03)=3C2*0.03^(2) *(1-0.03)^(3-2)`

`P(2;3;0.03)=3*0.0009*0.97 = 0.002619`

The probability when we replace x by 3 is:

`P(3;3;0.03)=3C3*0.03^(3) *(1-0.03)^(3-3)`

`P(3;3;0.03)=1*0.000027*1 = 0.000027`

So, the probability P that two or more orders are not shipped on time is:

P = P(2;3;0.03) + P(3;3;0.03)

P = 0.002619 + 0.000027

P = 0.002646