1. [20 Points] Assume two Boolean random variables A and B that can take on valu

Question

1. [20 Points] Assume two Boolean random variables A and B that can take on values True and False. Assume the probability P(A=True)-0.6 and P(B=True)-03. Give probability distributions for P(A) and P(B) Let's assume the variables A and B from part a. are independent. Give the joint probability distribution P(A,B) Let us assume the joint probability distribution of A and B is defined as: a. b. c. PA,B) True True False False True False rue False 0.2 0.4 0.3 Please calculate the following probabilities: PB-true) · rueB- P(B = true|A = false) ·

Explanation / Answer

a) here probability distribution of A :

P(A=true) =0.6

P(A=false) =0.4

probability distribution of B :

P(B=true) =0.3

P(B=false) =0.7

b) as for independence P(A,B) =P(A)*P(B) therefore below is joint probability table:

c)

P(B=true) =P(A=true;B=true)+ =P(A=false;B=true) =0.2+0.1 =0.3

P(A=true|B=false) =0.4/(0.4+0.3)=0.5714

P(B=true|A=false) =0.1/(0.1+0.3) =0.25

A B P(A,B) TRUE TRUE 0.18 TRUE FALSE 0.42 FALSE TRUE 0.12 FALSE FALSE 0.28

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