Question

2. (a) 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.

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

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

Let rotten oranges be denoted by 'R'

Probability that orange is rotten given it is supplied from A = P(R | A)

= 0.01 (1%)

Similarly P(R | B ) = 0.03

P(R | C) = 0.10

Probability of supplying from supplier A = P(A) = 0.70

Similarly P(B) = 0.15

P(C) = 0.15

i. Find the probability that a randomly selected orange from the box is rotten.

If it is rotten it can be from supplier A, B or C

P(Rotten) =

= P(R | A) P(A) + P(R | B) P(B) + P( R|C ) P(C)

= 0.01 * 0.70 + 0.03 * 0.15 + 0.10 * 0.15

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

We can model this as a binomial distribution

Where X is the event of being a rotten orange and the probability is 0.0265

P(X =x) =

P(More than 1 is rotten) = P( X > 1)

= 1 - P( X 1 )

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

= 1 - ()

= 1 - (0.874 + 0.119)

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

The probability of getting 1 rotten tomato out of 10 =

Where 10C1 = no. ways that 1 tomato is rotten eg. 1st, 2nd, ..10th

= probability that 1 is rotten

= probability that 9 are not rotten.

P( from supplier C | 1 rotten out of 10 ) =

Where the numerator considers probability from only supplier C and the denominator considers any supplier.

   = 0.015

P( from supplier C | 1 rotten out of 10 ) =