(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 13 balls in total, 5 white and 8 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 Xi given that X2 - 1 Px, x, (0l1)- (b) Calculate the conditional probability mass function X1 given that X2 - 0 Prix,(010) Px, x, (1 10)- For parts (c) and (d), assume that the balls are selected with replacement (c) Calculate the conditional probability mass function Xi given that X2 - 1 (d) Calculate the conditional probability mass function Xi given that X2-0

Explanation / Answer

a)

P(x1|X2)(0|1) =P(x1=0; x2=1)/P(x2=1)=(5/13)*(8/12)/(8/13)=5/12

P(x1|X2)(1|1) =(8/13)*(7/12)/(8/13)=7/12

b)

P(x1|X2)(0|0)=(5/13)*(4/12)/(5/13)=4/12=1/3

P(x1|X2)(1|0)=(8/13)*(5/12)/(5/13)=8/12=2/3

c)due to independence:

P(x1|X2)(0|1) =P(x1=0)=5/13

P(x1|X2)(1|1) =P(x1=1)=8/13

d)

due to independence:

P(x1|X2)(0|0) =P(x1=0)=5/13

P(x1|X2)(1|0) =P(x1=1)=8/13

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