Four balls are selected at random without replacement from an urn containing three white balls and five blue balls. Find the probability that two or three of the balls are white.

prob (two are white) = C(4,2)((3/8)^2(5/8)^2

Prob(three are white = C(4,3)(3/8)^3(5/8)

add them up.

To find the probability that two or three of the balls are white, we first need to calculate the total number of ways to select four balls from the urn.

The total number of balls in the urn is 3 white balls + 5 blue balls = 8 balls.

The total number of ways to select 4 balls from 8 is given by the combination formula (nCr), which is calculated as: n! / (r!(n-r)!)

In this case, the number of ways to select 4 balls from 8 is 8! / (4!(8-4)!).

Calculating this:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
4! = 4 x 3 x 2 x 1 = 24
(8-4)! = 4 x 3 x 2 x 1 = 24

So, 8! / (4!(8-4)!) = 40320 / (24 x 24) = 40320 / 576 = 70

Therefore, there are 70 different ways to select 4 balls from the urn.

Next, we need to find the number of ways to select two or three white balls from the urn.

Option 1: Selecting two white balls:
The number of ways to select two white balls is given by the combination of 2 white balls from the 3 available white balls: 3! / (2!(3-2)!).

Calculating this:
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
(3-2)! = 1

So, 3! / (2!(3-2)!) = 6 / (2 x 1 x 1) = 6 / 2 = 3

Option 2: Selecting three white balls:
The number of ways to select three white balls is given by the combination of 3 white balls from the 3 available white balls: 3! / (3!(3-3)!).

Calculating this:
3! = 3 x 2 x 1 = 6
3! = 3 x 2 x 1 = 6
(3-3)! = 0! = 1

So, 3! / (3!(3-3)!) = 6 / (6 x 1 x 1) = 6 / 6 = 1

Therefore, there are 3 different ways to select two white balls and 1 way to select three white balls.

Finally, we can calculate the probability that two or three of the balls are white by summing up the number of ways to select two or three white balls and dividing it by the total number of ways to select four balls:

(3 + 1) / 70 = 4 / 70 = 2 / 35

So, the probability that two or three of the balls are white is 2/35.