Sample space S is partitioned into E_1 and E_2 such that P(E_1) = 0.60 and P(E_2

Question

Sample space S is partitioned into E_1 and E_2 such that P(E_1) = 0.60 and P(E_2) = 0.40. Let D be an event such that P(D|E_1) = 0.3 and P(D|E_2) = 0.45. a) Find P(D) and P(D') b) Find P(E_1 | D) and P(E_1 | D'). c) Find P(E_2 | D) and P(E_2 | D'). a) P(D) = (Round to three decimal places as needed.) P(D') = (Round to three decimal places as needed. b) P(E_1 | D) = (Round to three decimal places as needed.) P(E_1 | D') = (Round to three decimal places as needed.) c) P(E_2 | D) = (Round to three decimal places as needed.) P(E_2|D') = (Round to three decimal places as needed.)

Explanation / Answer

a) P(D)=P(E1)*P(D|E1)+P(E2)*P(D|E2) =0.6*0.3+0.4*0.45=0.36

b)P(D') =1-P(D)=1-0.36=0.64

c)P(E1|D) =P(E1)*P(D|E1)/P(D) =0.5

d)P(E1|D')=P(E1)*P(D'|E1)/P(D') =0.6*0.7/0.64=0.6563

e)P(E2|D)=P(E2)*P(D|E2)/P(D2)=0.5

f)P(E2|D')= 1-0.6563 =0.3438

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