91Ó°ÊÓ

The possibility of a happening or outcome occurring supported by the occurrence of a preceding event or outcome is understood as a contingent probability.

The updated probability of the next or conditional, event is multiplied by the probability of the preceding, or conditional, event to get introduce contingent probability.

Prove:

$\frac{P(Hâˆ\pounds E)}{P(Gâˆ\pounds E)}=\frac{P\left(H\right)}{P\left(G\right)}×\frac{P(Eâˆ\pounds H)}{P(Eâˆ\pounds G)}$

$P\left(B\right)$isn't capable $0$for occurrences $A$and$B$

$P(Aâˆ\pounds B)=\frac{P(Aâˆ\copyright B)}{P\left(B\right)}$

Begin with the LHS and apply the definition:

$\frac{P\left(H\right|E)}{P\left(G\right|E)}=\frac{{\displaystyle \frac{P(Hâˆ\copyright E)}{P\left(E\right)}}}{{\displaystyle \frac{P(Gâˆ\copyright E)}{P\left(E\right)}}}$

$=\frac{P(Hâˆ\copyright E)}{P(Gâˆ\copyright E)}$

and on the RHS, using the identical definition:

$\frac{P\left(G\right)}{P\left(H\right)}×\frac{P\left(E\right|H)}{P\left(E\right|G)}=\frac{P\left(H\right)}{P\left(G\right)}×\frac{{\displaystyle \frac{P(Hâˆ\copyright E)}{P\left(H\right)}}}{{\displaystyle \frac{P(Gâˆ\copyright E)}{P\left(G\right)}}}$

=$=\frac{P(Hâˆ\copyright E)}{P(Gâˆ\copyright E)}$

As a result, it's been established.

The following statements are converted into mathematical equations.

$P\left(H\right)=3P\left(G\right)$

$P\left(E\right|G)=2P(E\left|H\right)$

Using the formula,

$$\begin{gathered} \frac{P\left(H\right|E)}{P\left(G\right|E)}=\frac{P\left(H\right)}{P\left(G\right)}×\frac{P\left(E\right|H)}{P\left(E\right|G)} \\ =\frac{P\left(H\right)}{P\left(G\right)}×\frac{P\left(E\right|H)}{P\left(E\right|G)} \end{gathered}$$

Replace the given facts together with your own.

localid="1649658537441" $$\begin{gathered} \frac{P\left(H\right|E)}{P\left(G\right|E)}=\frac{3P\left(G\right)}{P\left(G\right)}×\frac{P\left(E\right|H)}{2P\left(E\right|H)} \\ =3×\frac{1}{2} \\ =1.5 \end{gathered}$$

multiplying by $P\left(G\right|E)$and equating the primary and last expressions:

$P\left(H\right|E)=1.5×P(G\left|E\right)$$⇒P(Hâˆ\pounds E)>P(Gâˆ\pounds E)$