urn contains 4 white and 6 red rolls. Four balls are drawn at random (without replacement) from the urn. Find the probability distribution of number of white balls?

prob(white) = 4/10 = 2/5, prob(red) = 6/10 = 3/5

no white balls --- (3/5)^4 = 81/625
one white ball -- C(4,1) (2/5) (3/5)^3 = 216/625
two white balls -- C(4,2) (2/5)^2 (3/5)^2 = 216/625
three white balls -- C(4,3) (2/5)^3 (3/5) = 96/625
four white balls -- (2/5)^4 = 16/625

notice that the sum of these is 1 , as it should be

To find the probability distribution of the number of white balls drawn, we need to calculate the probabilities for each possible outcome.

First, let's determine the sample space, which represents all the possible outcomes when four balls are drawn from the urn. The number of ways to choose 4 balls from the total of 10 balls (4 white + 6 red) is given by the combination formula: C(10, 4) = 10! / (4! * (10 - 4)!) = 210.

Next, we'll calculate the probabilities for each number of white balls (0, 1, 2, 3, 4) being drawn.

1. Probability of drawing 0 white balls:
Selecting 4 balls from the 6 red balls: C(6, 4) = 6! / (4! * (6 - 4)!) = 15
P(0 white balls) = 15 / 210 = 1 / 14

2. Probability of drawing 1 white ball:
Selecting 1 white ball from the 4 white balls and 3 red balls from the remaining 6 red balls: C(4, 1) * C(6, 3) = 4 * 20 = 80
P(1 white ball) = 80 / 210 = 4 / 21

3. Probability of drawing 2 white balls:
Selecting 2 white balls from the 4 white balls and 2 red balls from the remaining 6 red balls: C(4, 2) * C(6, 2) = 6 * 15 = 90
P(2 white balls) = 90 / 210 = 3 / 7

4. Probability of drawing 3 white balls:
Selecting 3 white balls from the 4 white balls and 1 red ball from the remaining 6 red balls: C(4, 3) * C(6, 1) = 4 * 6 = 24
P(3 white balls) = 24 / 210 = 4 / 35

5. Probability of drawing 4 white balls:
Selecting all 4 white balls: C(4, 4) = 1
P(4 white balls) = 1 / 210 = 1 / 210

Hence, the probability distribution of the number of white balls drawn is as follows:

P(0 white balls) = 1/14
P(1 white ball) = 4/21
P(2 white balls) = 3/7
P(3 white balls) = 4/35
P(4 white balls) = 1/210