Let `E_(1) and E_(2)` be two independent events such that `P(E_(1))=P_(1)` and `P(E_(2))=P_(2)`, describe in words of the events whose probabilities are
(i) `P_(1)P_(2)` (ii) `(1-P_(1))P_(2)`
(iii) `1-(1-P_(1))(1-P_(2))` (iv) `P_(1)+P_(2)-2P_(1)P_(2)`

Explanation

`P(E_(1))=P_(1)and P(E_(2))=P_(2)`
(i) `P_(1)P_(2)rArrP(E_(1))cdotP(E_(2))=P(E_(1)capE_(2))`
So, `E_(1) and E_(2)` occur.
(ii) `(1-P_(1))P_(2)=P(E_(1))'cdotP(E_(2))=P(E_(1)'capE_(2))`
(iii) `1-(1-P_(1))(1-P_(2))=1-P(E_(1))'P(E_(2))'=1-P(E_(1)capE_(2)')`
`=1-[1-P(E_(1)cupE_(2))]=P(E_(1)cupE_(2))`
So, either `E_(1)orE_(2)` or both `E_(1) and E_(2)` occurs.
(iv) `P_(1)+P_(2)-2P_(1)P_(2)=P(E_(1))+P(E_(2))-2P(E_(1))cdotP(E_(2))`
`=P(E_(1))+P(E_(2))-2P(E_(1))cap(E_(2))`
`=P(E_(1)cupE_(2))-P(E_(1))capE_(2))`
So, either `E_(1) or E_(2)` occurs but not both.