If `E_(1) and E_(2)` are two events such that `P(E_(1)) = 0.5, P(E_(2)) = 0.3 and P(E_(1) and E_(2)) = 0.1`, find (i) `P(E_(1) " or " E_(2))` (ii) `P(E_(1) " but not " E_(2))` (iii) `P(E_(2) " but not " E_(1))` (iv) `P(" neither " E_(1) " nor " E_(2))`

If `E_(1) and E_(2)` are two events such that `P(E_(1)) = 0.5, P(E_(2)

1.	If `E_(1) and E_(2)` are two events such that `P(E_(1)) = 0.5, P(E_(2)) = 0.3 and P(E_(1) and E_(2)) = 0.1`, find (i) `P(E_(1) " or " E_(2))` (ii) `P(E_(1) " but not " E_(2))` (iii) `P(E_(2) " but not " E_(1))` (iv) `P(" neither " E_(1) " nor " E_(2))`
Answer» We have `P(E_(1)) = 0.5, P(E_(2)) = 0.3 and P(E_(1) nn E_(2)) = P(E_(1) and E_(2)) = 0.1`. `therefore P(bar(E_(1))) = {1 - P(E_(1))} = (1 - 0.5) = 0.5,` and `P(bar(E_(2))) = {1 - P(E_(2))} = (1 - 0.3) = 0.7.` Thus, we have (i) `P(E_(1) " or " E_(2)) = P(E_(1) uu E_(2))` `= P(E_(1)) + P(E_(2)) - P(E_(1) nn E_(2))` `= (0.5 + 0.3 - 0.1) = 0.7.` (ii) `P(E_(1) " but not " E_(2)) = P(E_(1) nn bar(E_(2)))` `= P(E_(1)) - P(E_(1) nn E_(2))` `= (0.5 - 0.1) = 0.4.` (iii) `P(E_(2) " but not " E_(1)) = P(E_(2) nn bar(E_(1)))` `= P(E_(2)) - P(E_(2) nn E_(1))` `= P(E_(2)) - P(E_(1) nn E_(2)) = (0.3 - 0.1) = 0.2.` (iv) `P("neither " E_(1) " nor " E_(2)) = P(" not " E_(1) " and not " E_(2))` `= P(bar(E_(1)) and bar(E_(2))) = P(bar(E_(1)) nn bar(E_(2)))` `= P(bar(E_(1) uu E_(2))) = 1 - P(E_(1) uu E_(2))` `= 1 - P(E_(1) uu E_(2))` `= (1 - 0.7) = 0.3` [using (i) ].

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If `E_(1) and E_(2)` are two events such that `P(E_(1)) = 0.5, P(E_(2)) = 0.3 and P(E_(1) and E_(2)) = 0.1`, find (i) `P(E_(1) " or " E_(2))` (ii) `P(E_(1) " but not " E_(2))` (iii) `P(E_(2) " but not " E_(1))` (iv) `P(" neither " E_(1) " nor " E_(2))`

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