Five 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.

To find the probability that two or three of the balls are white, we need to consider two cases: when two balls are white and when three balls are white.

Case 1: Two balls are white:
To calculate this probability, we need to calculate the probability of choosing two white balls and three blue balls.

First, let's determine the total number of ways to choose five balls from the urn:
Total number of ways = (Number of white balls + Number of blue balls) choose (Number of balls selected)
Total number of ways = (3 + 5) choose 5 = 8 choose 5 = 56

Next, we calculate the number of ways to choose two white balls and three blue balls:
Number of ways = (Number of white balls choose 2) * (Number of blue balls choose 3)
Number of ways = (3 choose 2) * (5 choose 3) = 3 * 10 = 30

So, the probability of selecting two white balls is:
Probability = Number of ways / Total number of ways = 30 / 56 = 15 / 28

Case 2: Three balls are white:
Similarly, we need to calculate the probability of choosing three white balls and two blue balls.

Number of ways = (Number of white balls choose 3) * (Number of blue balls choose 2)
Number of ways = (3 choose 3) * (5 choose 2) = 1 * 10 = 10

So, the probability of selecting three white balls is:
Probability = Number of ways / Total number of ways = 10 / 56 = 5 / 28

Now, to find the total probability, we add the probabilities of the two cases together:
Total probability = Probability of two white balls + Probability of three white balls
Total probability = 15 / 28 + 5 / 28 = 20 / 28 = 5 / 7

Therefore, the probability that two or three of the balls are white is 5/7.