For the three events `A`, `B` and `C`, `P`(exactly one of the events A or B occurs) `= P`(exactly one of the events B or C occurs) `=P`(exactly one of the events C or A occurs)`= p` and `P`(all the three events occur simultaneously) `= p^2`, where `0ltplt1/2`. Then, find the probability of occurrence of at least one of the events `A`, `B` and `C`.

Explanation

`P`( exactly one of `A` or `B`)`=P(A cup B)-P(A cap B)=frac{1}{4}=P(A)+P(B)-2 P(A cap B)`

`P` (exactly one of `C` or `B`)`=P(C cup B)-P(C cap B)=frac{1}{4}=P(C)+P(B)-2 P(C cap B)`

`P` (exactly one of `A` or `C`)`=P(A cup C)-P(A cap C)=frac{1}{4}=P(A)+P(C)-2 P(A cap C)`

Adding all

`2 P(A)+2 P(B)+2 {P}(c)-2 {P}(A cap B)-2 P(A cap C)-2 P(B cap C)=frac{3}{4}`

`P(A)+P(B)+P(c)-P(A cap B)-P(A cap C)-P(B cap C)=frac{3}{8}`

`P(A cup B cup C)=P(A)+P(B)+P(C)-P(A cap B)-P(A cap C)-P(B cap C)+` `P(A cap B cap C)`

`=frac{3}{8}+frac{1}{16}=frac{7}{16}`