# What is the probability that least 1 box is damaged and at most 1 box is damaged in the following case:A hypermarket received a delivery of 10000 boxes of apples where 100 of the boxes are damaged....

What is the probability that least 1 box is damaged and at most 1 box is damaged in the following case:

A hypermarket received a delivery of 10000 boxes of apples where 100 of the boxes are damaged. A sample of 10 randomly chosen boxes is picked.

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### 1 Answer

The hypermarket received a delivery of 10000 boxes and 100 of the boxes are damaged. A sample of 10 boxes is picked.

There are C(10000, 10) ways of picking 10 boxes out of the 10000.

The probability that at most one of the boxes is damaged is `(9900C9*100C1)/(10000C10) ~~ 0.0914`

The number of ways of picking boxes so that at least one of them is damaged is 9900C0*100C10 + 9900C1*100C9 + 9900C2*100C8 + 9900C3*100C7 + 9900C4*100C6 + 9900C5*100C5 + 9900C6*100C4 + 9900C7*100C3 + 9900C8*100C2 + 9900C9*100C1. This gives the probability that at least one of the boxes is damaged as : (9900C0*100C10 + 9900C1*100C9 + 9900C2*100C8 + 9900C3*100C7 + 9900C4*100C6 + 9900C5*100C5 + 9900C6*100C4 + 9900C7*100C3 + 9900C8*100C2 + 9900C9*100C1)/(10000C10). This is also equal to `1 - 0.0914 ~~ 0.9085`