Let E and F be events such that Pr (E or F) = .95 and Pr (E) = .37.  What is Pr (F) if E and F are independent?



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Addition theorem of probability for two events states that “If E and F are two events associated with a random experiment, then Pr(E `uu` F) =Pr(E) + Pr(F) - Pr(E `nn` F).”

Here E and F are two independent events. So, Pr(E `nn` F)= Pr(E)* Pr(F)

Hence, Pr(E ` ` F) =Pr(E) + Pr(F) - Pr(E)*Pr(F).

Here, Pr (E `uu` F) =0 .95 and Pr (E) = 0.37.

Plugging in the values in the relation yields:

0.95=0.37+ Pr(F) - 0.37*Pr(F)

`rArr` Pr(F)[1-0.37]=0.95-0.37 =0.58

`rArr ` Pr(F)=0.58/0.63=0.920635

Therefore, Pr(F)=0.92, if E and F are independent events.


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