P(F1)= Prob of letter in folder 1 = 1/3
P(F2)= PRob of letter in folder 2 =1/3
P(F3)= Prob of letter in folder 3 =1/3
P(not found in F1)= prob of letter not found in folder F1 = 1-a
P( present in F1)= Prob of letter prsent in F1
question asked = P(letter present in F1 | letter not found in F1) = P(letter not found in F1 | letter present in F1) * P(letter present in F1) / P(letter not found in F1 | letter present in F1) * P(letter present in F1) + P(letter not found in F1 | letter present in F2) * P(letter present in F2) + P(letter not found in F1 | letter present in F3) * P(letter present in F3)
=>P(letter not found in F1 | letter present in F1) * P(letter present in F1) = (1-a) * 1/3
=> P(letter not found in F1 | letter present in F2) * P(letter present in F2) = 1 * 1/3 as if letter is present in F2 then prob of not finding it in F1 will be defnititly 1.
=> P(letter not found in F1 | letter present in F3) * P(letter present in F3) = 1 * 1/3
so,[(1-a) * 1/3] /[ (1-a) * 1/3 + 1/3 + 1/3] = 1-a / 3-a
please correct me