(1 point) Consider another ubiquitous probability-course urn containing well-mix

Question

(1 point) Consider another ubiquitous probability-course urn containing well-mixed black and white balls. There are 19 balls in total, 7 white and 12 black. 2 are chosen, one at a time and at random. Let X, be 1 if the ith ball selected is black, and 0 otherwise For parts (a) and (b), assume that the balls are selected without replacement (a) Calculate the conditional probability mass function X, given that X2 = 1 PXlXe (011) 7/18 PX1X2 (111)= 11/18 (b) Calculate the conditional probability mass function X, given that X2 = 0 2x1x2 (010) = PXix, (110) = For parts (c) and (d), assume that the balls are selected with replacement (c) Calculate the conditional probability mass function X, given that X2 = 1 Pax, (011) = 2x1x2 (111) = (d) Calculate the conditional probability mass function X1 given that X20 Park, (010) = 7x1x2 (110) =

Explanation / Answer

a)

Px2(1) =first white second black+first black second black =(7/19)*(12/18)+(12/19)*(11/18) =12/19

P(0|1) =(7/19)*(12/18)/(12/19) =7/18

P(1|1) =(12/19)*(11/18)/(12/19)=11/18

b)

Px2(0) =white white+black white =(7/19)*(6/18)+(12/19)*(7/18)=7/19

P(0|0) =(7/19)*(6/18)/(7/19)=6/18

P(1|0)=(12/19)*(7/18)/(7/19)=12/18

c)

P(1)=first white second black+first black second black =(7/19)*(12/19)+(12/19)*(12/19) =12/19

P(0|1)= 7/19

P(1|1)=12/19

d)

Px2(0) =white white+black white =(7/19)*(7/19)+(12/19)*(7/19)=7/19

P(0|0)=7/19

P(1|0)=12/19

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