could someone show me step by step how to do part C? 2. Box A contains 6 red bal

Question

could someone show me step by step how to do part C?

2. Box A contains 6 red balls and 4 green balls. Three of these balls are selected at random (without replacement) and placed in box B, which was originally empty. One ball is then selected at random from box B. Let X denote the number of red balls among the three balls transferred from A to B. a) Find the probability mass function for X b) Find the probability that the ball selected from box B is red. c) Given that the ball selected from box B is red, find the conditional probabilities that X 1, X 2, and X 3. That is, for each of the possibilities 1, 2, and 3, find the probability that exactly this many red balls were transferred from box A to box B given that the ball selected from box B is red.

Explanation / Answer

(a)

Distribution of X

(b)P(R) =P(X=1).P(R | X=1) + P(X=2).P(R | X=2) + P(X=3).P(R |X=3)

= (3/10).(1/3) + (1/2)(2/3) + (1/6)(3/3)

= 1/10 + 1/3 + 1/6

= 3/5

(c)the conditional probablity that the selected ball from B is red and X=1

= P(X=1 | R)

= [P(X=1).P(R | X=1)] / P(R)

=[(3/10)(1/3)] / (3/5)

= 1/6

the conditional probablity that the selected ball from B is red and X=2

= P(X=2 | R)

= [P(X=2).P(R | X=2)] / P(R)

= [ (1/2)(2/3)] / (3/5)

= 5/9

the conditional probablity that the selected ball from B is red and X=2

= P(X=3 | R)

= [P(X=3).P(R | X=3)] / P(R)

= [(1/6)(3/3)] / (3/5)

= 5/18

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