Conditional Probability

Question

Assume that Pr[E]=1/2 , Pr[E|F]=2/5, and Pr[E’ intersectionF’]=4/15

1) find Pr[F]

2) find Pr[E intersection F]

Answer 1

First use Pr[E’ ∩ F’]=4/15 and DeMorgan's Law that E’ ∩ F' = (E ∪ F)' to conclude that Pr[ (E ∪ F)' ] = 4/15.

So Pr[E ∪ F] = 11/15.

We also know that Pr[E ∪ F] = Pr[E] + Pr[F] - Pr[E ∩ F].

Since we are given Pr[E] = 1/2, if we can find an expression for Pr[E ∩ F] in terms of Pr[F], our only unknown will be Pr[F].

And we can find this, since Pr[E | F] = 2/5 and by definition of conditional probability,

P[E | F] = Pr[E ∩ F] / P[F],

so we know 2/5 = Pr[E ∩ F] / Pr[F]

and then, multiplying both sides of by P[F], gives

2/5 P[F] = Pr[E ∩ F]

Now to find Pr[F], we can substitute into Pr[E ∪ F] = Pr[E] + Pr[F] - Pr[E ∩ F].

Substituting, we have

11/15 = 1/2 + Pr[F] - 2/5 Pr[F]

11/15 = 1/2 + 3/5 Pr[F]

To clear fractions, multiply both sides by 30 to get

22 = 15 + 18 Pr[F]

7 = 18 Pr[F]

7/18 = Pr[F]

Finally, to find Pr[E ∩ F], we substitute back into

2/5 P[F] = Pr[E ∩ F] to get

2/5 * 7/18 = Pr[E ∩ F]

which becomes

7/45 = Pr[E ∩ F]