Question

Please answer with clear steps -

The owner of a juice bar, Sam, orders oranges from 3 suppliers, namely A, B and C. From past record, each box of oranges from suppliers A, B, C contains 1%, 3% and 10% rotten oranges respectively. Assume that Sam orders 70% of the oranges from supplier A, 15% of the oranges from supplier B and 15% of the oranges from supplier C. One morning, a box of oranges arrives but Sam does not know which supplier sent it. A random check is then conducted to see if the oranges are rotten.

i. Find the probability that a randomly selected orange from the box is rotten. [2 marks]

ii. Find the probability that there are more than 1 rotten orange among 5 randomly selected oranges. [3 marks]

iii. Sam inspects 10 oranges, and finds that 1 is rotten. With this information, find the probability that the box comes from supplier C.

Answer 1

(i)

A randomly selected orange from the box is rotten is given by:

(0.70 X 0.01) + (0.15 X 0.03) + (0.15 X 0.10)

= 0.007 + 0.0045 + 0.015

= 0.0265

So,

Answer is:

0.0265

(ii)

Binomial Distribution

n =5

p = 0.0265

q = 1 - p = 0.9735

So,

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

So,

P(X>1)= 1 - 0.9933

= 0.0067

So,

Answer is:

0.0067

(iii)

Given:
P(A) = 0.70

P(B) = 0.15

P(C) =0.15

P(R/A) = 0.01

P(R/B) = 0.03

P(R/C) = 0.10

By Bayes Theorem, we have:

= 0.5660

So

Answer is:

0.5660