Question

a box of oranges is inspected by examining three randomly selected oranges drawn in replacement. If all the three oranges are good, the box is approved for sale,otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 bad ones will be approved for sale

Answer 1

$\frac{64}{125}$

Answer 2

Let G denote a good orange and B denote a bad orange.

Total number of oranges in the box = 15 Number of good oranges = 12 Number of bad oranges = 3

The inspection process involves drawing three oranges with replacement. This means that after each orange is drawn and its quality is noted, it is put back into the box. Consequently, the probability of drawing a good or bad orange remains constant for each draw, and the draws are independent events.

For the box to be approved for sale, all three randomly selected oranges must be good.

1. Probability of drawing a good orange in a single draw:

   $P(\text{Good}) = \frac{\text{Number of good oranges}}{\text{Total number of oranges}}$

   $P(\text{Good}) = \frac{12}{15} = \frac{4}{5}$

2. Probability of drawing three good oranges in a row with replacement:

   Since each draw is independent:

   $P(\text{1st orange is good}) = \frac{4}{5}$

   $P(\text{2nd orange is good}) = \frac{4}{5}$ (because the first orange was replaced)

   $P(\text{3rd orange is good}) = \frac{4}{5}$ (because the second orange was replaced)

   The probability that all three oranges drawn are good is the product of their individual probabilities:

   $P(\text{Approved}) = P(\text{1st good}) \times P(\text{2nd good}) \times P(\text{3rd good})$

   $P(\text{Approved}) = \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5}$

   $P(\text{Approved}) = \frac{4^3}{5^3}$

   $P(\text{Approved}) = \frac{64}{125}$

The probability that the box will be approved for sale is $\frac{64}{125}$.