Question

There are 35 identically sized balls in a box, where 5 of each of the colors are present: red, orange, yellow, green, blue, indigo, and violet. Justify your answer for each.

a. Suppose you draw a sample of seven balls without replacement from the box. What is the probability that the sample contains balls of exactly two colors? (for example, the event for which you have 4 red and 3 blue would satisfy)

b. Suppose you draw a sample of seven balls with replacement from the box (you draw one, then replace, draw one, then replace, and so on). What is the probability that the sample contains balls of exactly two colors?

c. Suppose you draw a sample of three balls without replacement. Define ? to be the random variable on the number of red balls in the sample and ? the random variable on the number of orange balls in the sample. Find the joint probability distribution function ??,?(?, ?) for the random variables ? and

Answer 1

a. there are total 35 balls. 5 of each colour

No. of ways of selecting any 7 of the 35 = 35c7

No. of ways to select 7 balls of exactly 2 colours out of those 35 can be done as follows:

1. You select 2 of those 7 colours, this can be done in 7c2 ways

2. Now we have a total of 10 balls, 5 of each colour, and we select 7. Suppose we have red and orange the distribution can be 5R 2O, 4R 3O, 3R 4O, 2R 5O.

Therefore total = (5c5 * 5c2 + 5c4 * 5c3 + 5c3 * 5c4 + 5c2 * 5c5) * 7c2 = 2520

Probability = 2520 / (35c7) = 0.00037

b. With replacement the probability of picking a ball of a particular = 5/35 = 1/7

Like in the previous case if we take red and orange balls, 5R 2O, 4R 3O, 3R 4O, 2R 5O.

5R 2O probability = (1/7)^5 * (1/7)^2 [(1/7) will be the probability of doing it every time ]

Therefore ( 5R 2O, 4R 3O, 3R 4O, 2R 5O) probability = (1/7) ^ 7 * 4

and this can be done with 2 colours of the 7 so 7c2 ways

Total probability = 7c2 * (1/7)^7 *4 = 0.0001

c. X: red balls in sample and Y : orange balls in sample

X	Y	Probability
0	3	5c3/ 35c3
1	2	(5c1 *5c2) / 35c3
2	1	(5c1 *5c2) / 35c3
3	0	5c3/ 35c3