In a Spam filter design, following events are defined,

s: email is spam A1=email contains string "cheapest"

A2=email contains string "lottery" A3=email contains string "credit"

it is found,

P(S) =0.2

P(A1|S)=0.2 P(A2|S)=0.4 P(A3|S)=0.2

P(A1|S')=0.05 P(A2|S')=0.1 P(A3|S')=0.01

conditional on S, assume A1 , A2 , A3 are independent.

find the probability that email is spam if,

1)A1 occurs (p1) 2) both A1 and A2 occur (p2)

3)A1,A2,A3 occur (p3) 4)explain why p3>p2>p1

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i solved 1st one using Bayes theorem and i got 0.5 as the answer.

i need help for 2nd ans 3rd parts.

i couldn't get an answer for 2nd part, I got 0.25 when i assumed (s Ʌ A1) and (S Ʌ A2) which must be wrong because it should be greater than 0.5

by the way is it wrong to assume (s Ʌ A1) and (S Ʌ A2) are independent when A1 and A2 are independent?