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
To find the probability that two or three of the balls are white, we need to consider two cases:
Case 1: exactly two balls are white.
In this case, we need to choose two white balls and two non-white balls from the urn. The number of ways to choose two white balls from three white balls is the combination "3 choose 2" (denoted as C(3, 2)). Similarly, the number of ways to choose two non-white balls from five non-white balls is "5 choose 2" (denoted as C(5, 2)). Therefore, the total number of favorable outcomes for this case is C(3, 2) * C(5, 2).
Case 2: exactly three balls are white.
In this case, we need to choose three white balls and one non-white ball from the urn. The number of ways to choose three white balls from three white balls is C(3, 3), which is 1 since there is only one way to choose all three white balls. The number of ways to choose one non-white ball from five non-white balls is C(5, 1), which is 5. Therefore, the total number of favorable outcomes for this case is C(3, 3) * C(5, 1).
Next, we need to calculate the total number of possible outcomes, which is the number of ways to choose four balls from the total of eight balls in the urn. This can be calculated as C(8, 4).
Finally, we can calculate the probability by dividing the sum of the favorable outcomes from both cases by the total number of possible outcomes:
Probability = (C(3, 2) * C(5, 2) + C(3, 3) * C(5, 1)) / C(8, 4).
Now let's calculate the probability.