For n independent events `A_i's, p(A_i)=1/(1+i),i=1,2,...n`. The probability that at least one of the events occurs, is

Options

A. `(1)/(4)`
B. `(2)/(3)`
C. `(3)/(4)`
D. `(1)/(24)`

Answer

C

Explanation

`P(A_(1)) = (1)/(1+1) = (1)/(2)`
`P(A_(2)) = (1)/(3)`
`P(A_(3)) = (1)/(4)`
`therefore` Probability that at leat one of these events occur is `P(A_(1) uu A_(2) uu A_(3)) = P(A_(1)) +P(A_(2)) +P(A_(3))` are independent events.
`P(A_(1) uu A_(2)uuA_(3)) = P(A_(1)) +P(A_(2)) +P(A_(3)) - P(A_(1) nn A_(2)) -P(A_(1) nn A_(3))`
` - P(A_(2) nn A_(3)) +P(A_(1) nn A_(2) nn A_(3))`
` = (1)/(2) +(1)/(3) +(1)/(4) -((1)/(2) xx (1)/(3)) -((1)/(2) xx (1)/(4)) - ((1)/(3) xx (1)/(4)) +((1)/(2) xx (1)/(3) xx (1)/(4)) = (3)/(4)`