Question

G and H are mutually exclusive events. $P\left(G\right)=0.5P\left(H\right)=0.3$

a. Explain why the following statement MUST be false: $P\left(H\right|G)=0.4$.

b. Find $P(HORG)$.

c. Are G and H independent or dependent events? Explain in a complete sentence

Short answer

(a) $P\left(G\text{and}H\right)=0$as G and H are mutually exclusive events, hence the statement $P\left(H|G\right)=0.4$must be false.

(b) localid="1648908239487" $P\left(HORG\right)=0.8$.

(c) G and H are dependent events.

Step-by-step solution

Given information (part a)

G and H are mutually exclusive events.$P\left(G\right)=0.5\text{and}P\left(H\right)=0.3$

Explanation (part a)

Given that $P\left(G\right)=0.5\text{and}P\left(H\right)=0.3$

We need to calculate $P\left(H|G\right)$

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

Also, G and H are mutually exclusive events,

$P\left(GandH\right)=0$

Thus, we have

localid="1649420858160" $$\begin{gathered} P\left(G|H\right)=\frac{0}{0.3} \\ P\left(G|H\right)=0 \end{gathered}$$

Given information (part b)

G and H are mutually events.$P\left(G\right)=0.5\text{and}P\left(H\right)=0.3$

Explanation (part b)

$P\left(G\right)=0.5\text{and}P\left(H\right)=0.3$

Also, GandH are mutually exclusive events,

$P\left(G\text{and}H\right)=0$

Thus, $P\left(H\text{or}G\right)$is calculated as

$$\begin{gathered} P\left(HorG\right)=P\left(H\right)+P\left(G\right)-P\left(H\text{and}G\right) \\ P\left(HorG\right)=0.3+0.5-0 \\ P\left(HorG\right)=0.8 \end{gathered}$$

Given information (part c)

G and H are mutually exclusive events.$P\left(G\right)=0.5\text{and}P\left(H\right)=0.3$

Explanation (part c)

Given that G and H are mutually exclusive events,

$P\left(G\text{and}H\right)=0$

The information that one event has occurred completely determines the probability of the other event, which is zero in this case. Thus, both events cannot occur at the same time, and therefore the events are dependent on each other.