A box contains 5 red bowls, 6 yellow bowls and 3 white bowls. If 4 bowls are randomly drawn from the box at the same time,
(a)find the probability that exactly 2 red bowls are drawn;
(b)find the probability that at least 2 red bowls are drawn.
To solve these probability problems, we need to first determine the total number of possible outcomes and the number of favorable outcomes.
(a) Probability of drawing exactly 2 red bowls:
Step 1: Calculate the total number of possible outcomes.
Since we are drawing 4 bowls at the same time, the total number of possible outcomes is determined by the formula for combinations:
nCr = n! / (r!(n-r)!)
In this case, n is the total number of bowls in the box (14) and r is the number of bowls drawn (4).
So, the total number of possible outcomes is 14C4 = 14! / (4!(14-4)!).
Step 2: Calculate the number of favorable outcomes.
We want to draw exactly 2 red bowls, so we need to consider the combinations of choosing 2 red bowls out of the 5 available, and 2 non-red bowls out of the remaining 9 bowls.
The number of favorable outcomes is given by 5C2 * 9C2 = 5!/(2!(5-2)!) * 9!/(2!(9-2)!).
Step 3: Calculate the probability.
The probability of exactly 2 red bowls is given by the number of favorable outcomes divided by the total number of possible outcomes:
P(exactly 2 red bowls) = (5C2 * 9C2) / (14C4).
(b) Probability of drawing at least 2 red bowls:
Step 1: Calculate the total number of possible outcomes (same as in part a).
Step 2: Calculate the number of favorable outcomes.
We want to draw at least 2 red bowls, so we need to consider the combinations of choosing 2, 3, or 4 red bowls out of the 5 available, and the remaining bowls from the non-red bowls.
The number of favorable outcomes is given by the sum of the combinations:
(5C2 * 9C2) + (5C3 * 9C1) + (5C4 * 9C0) = 5!/(2!(5-2)!) * 9!/(2!(9-2)!) + 5!/(3!(5-3)!) * 9!/(1!(9-1)!) + 5!/(4!(5-4)!) * 9!/(0!(9-0)!).
Step 3: Calculate the probability.
The probability of drawing at least 2 red bowls is given by the number of favorable outcomes divided by the total number of possible outcomes:
P(at least 2 red bowls) = [(5C2 * 9C2) + (5C3 * 9C1) + (5C4 * 9C0)] / (14C4).