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For the three events "A","B","andC","P"(...

For the three events `"A","B","andC","P"("e x a c t l yo n eo ft h ee v e n t sAorBo c c u r s")="P"("e x a c t l yo n eo ft h ee v e n t sBorCo c c u r s")="P"("e x a c t l yo n eo ft h ee v e n t sCandAo c c u r s")="p"` and`"P"("a l lt h et h r e ee v e n t so c c u rs i m u l t a n e o u s l y")="p"^2,"w h e r e"0<"p"<1//2.` Then, find the probability of occurrence of at least one of the three events `A ,B ,a n dCdot`

A

`(3p+2p^(2))/(2)`

B

`(p+3p^(2))/(4)`

C

`(p+3p^(2))/(2)`

D

`(3p+2p^(2))/(4)`

Text Solution

Verified by Experts

The correct Answer is:
A
We know that,
P(exactly one of A or B occurs)
`P(A)+P(B)-2P(A nn B)`
`:. P(A)+P(B)-2P(A nn B)=p " "`….(i)
Similarly, `P(B)+P(B)-2P(B nn C)=p" "`….(ii)
and `P(C )+P(A)-2(C nn A)=p" "`....(iii)
On adding Eqs. (i),(ii) and (iii), we get
`2[P(A)+P(B)+P(C )-P(A nn B)-P(B nnC)-P(C nn A)]=3p`
`rArr P(A)+P(B)+P(C )-P(A nn B)-P(B nn C)-P(C nn A)=(3p)/(2)" "`....(v)
It also given that, `P(A nn B nn C)=p^(2) " "`....(vi)
`:. ` P(at least one of the events A,B and C occurs)
`=P(A)+P(B)+P(C )-P(A nn B)-P(B nn C)-P(C nn A)+(Ann B nn C)`
`=(3p)/(2)+p^(2) " "` [from Eqs. (iv) and (v)]
`=(3p+2p^(2))/(2)`
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