Question

1. SAMPLING WITHOUT REPLACEMENT

a. A bin contains 4 red balls and 8 blue balls. Several balls are drawn from the bin. What is the probability that the 6th ball and the 8th ball drawn are both red?

b. A bin contains 5 red balls and 9 blue balls. Several balls are drawn from the bin. Given that the 4th ball drawn is blue, what is the probability that the 8th ball drawn is red?

c. A bin contains 5 red balls and 9 blue balls. Several balls are drawn from the bin. Given that the first ball drawn is blue, what is the probability that the second ball drawn is blue?

Could someone please explain how to do these? I get some right but others wrong and I have no idea why.

Answer 1

a) Total = 4 Red + 8 blue = 12

The 6th and 8th place are fixed as both have to be red. We can have different possible arrangements where there are only 2 red balls, 3 red balls and 4 red balls.

Different Possibilities:

When there are only 2 red balls. There is only 1 arrangement. that is the balls in the 6th and the 8th place have to be red the others are all blue.

#	1	2	3	4	5	6	7	8
Colour	B	B	B	B	B	R	B	R
Probability	8/12	7/11	6/10	5/9	4/8	4/7	3/6	3/5

Therefore the probability = (8/12) * (7/11) * (6/10) * (5/9) * (4/8) * (4/7) * (3/6) * (3/5) = 241920/19958400

When there are 3 Red balls. 2 places are already fixed. Now the 3rd red ball can move in any of the remaing 6 places in ⁶C₁ = 6 ways. The probability for each of the arrangements will be the same, hence we need to find the probability of 1 arrangement and multiply it by 6. Lets take for eg. Red in the 1st, 6th and 8th places

#	1	2	3	4	5	6	7	8
Colour	R	B	B	B	B	R	B	R
Probability	4/12	8/11	7/10	6/9	5/8	3/7	4/6	2/5

Therefore the probability of 1 arrangement having 3 red balls = (4/12) * (8/11) * (7/10) * (6/9) * (5/8) * (3/7) * (4/6) * (2/5) = 161280/19958400

Therefore for 6 arrangements = 6 * (161280/19958400) = 967680/19958400

When there are 4 Red balls. 2 places are already fixed. Now the 3rd and 4th red ball can move in any of the remaing 6 places in ⁶C₂ = 15 ways. We need to find the probability of 1 arrangement and multiply it by 15. Lets take for eg. Red in the 1st, 2nd, 6th and 8th places.

#	1	2	3	4	5	6	7	8
Colour	R	R	B	B	B	R	B	R
Probability	4/12	3/11	8/10	7/9	6/8	2/7	5/6	1/5

Therefore the probability of 1 arrangement having 4 red balls = (4/12) * (3/11) * (8/10) * (7/9) * (6/8) * (2/7) * (5/6) * (1/5) = 40320/19958400

Therefore for 15 arrangements = 15 * (40320/19958400) = 604800/19958400

Therefore the required probability = P(2 red Balls) + P(3 red balls) + P(6 red balls)

= 241920/19958400 + 967680/19958400 + 604800/19958400 = 1814400/19958400 = 0.0909

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