A box has 9 marbles, 4 of which are white and 5 of which are red. A sample of 5 marbles is selected randomly from the box without replacement.
What is the probability that exactly 3 are white and 2 are red? (Enter your answer as an exact fraction.)
prob(white) = 4/9
prob(red) = 5/9
prob(3 of 5 are white)
= C(5,3) (4/9)^3 (5/9)^2
= 10(64/729)(25/81)
= 16000/59049
To find the probability that exactly 3 marbles are white and 2 are red, we need to calculate the number of favorable outcomes (where 3 out of the 5 marbles selected are white and 2 are red) and divide it by the total number of possible outcomes.
First, let's calculate the total number of possible outcomes. From a box with 9 marbles, selecting 5 marbles without replacement means we are choosing 5 marbles from the 9 available. This can be calculated using a combination formula:
C(n, r) = n! / (r!(n - r)!)
In this case, n = 9 (total number of marbles in the box) and r = 5 (number of marbles being selected). Substituting the values, we have:
C(9, 5) = 9! / (5!(9 - 5)!)
= 9! / (5!4!)
= (9 * 8 * 7 * 6 * 5!) / (5! * 4 * 3 * 2 * 1)
= (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1)
= 3024 / 24
= 126
There are 126 possible outcomes when selecting 5 marbles from a box of 9 marbles.
Next, let's calculate the number of favorable outcomes (where exactly 3 are white and 2 are red). To do this, we need to calculate the number of ways to select 3 white marbles from the 4 available white marbles, and the number of ways to select 2 red marbles from the 5 available red marbles.
The number of ways to select 3 white marbles from 4 is given by:
C(4, 3) = 4! / (3!(4 - 3)!)
= 4! / (3! * 1!)
= (4 * 3!) / (3! * 1)
= 4
The number of ways to select 2 red marbles from 5 is given by:
C(5, 2) = 5! / (2!(5 - 2)!)
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / (2 * 1)
= 10
Therefore, there are 4 ways to select 3 white marbles and 10 ways to select 2 red marbles.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (4 * 10) / 126
= 40 / 126
= 20 / 63
So, the probability that exactly 3 marbles are white and 2 are red is 20/63.