Question

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`<br>A. `(3p+2p^(2))/(2)` B. `(p+3p^(2))/(4)` C. `(p+3p^(2))/(2)` D. `(3p+2p^(2))/(4)` Select the correct answer from above options

Answer 1

Correct Answer - 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)`