Question

Consider the following scenario:

Let $P\left(C\right)=0.4.$Let $P\left(D\right)=0.5$. Let $P\left(C|D\right)=0.6$.

a. Find P(C AND D).

b. Are C andD mutually exclusive? Why or why not?

c. Are C and D independent events? Why or why not?

d. Find P(C OR D).

e. Find P(D|C).

Short answer

(a) $P\left(CANDD\right)=0.3$.

(b) C and D are not mutually exclusive events.

(c)Cand D are not independent events.

(d) $P\left(CorD\right)=0.6$.

(e)$P\left(D|C\right)=0.75$

Step-by-step solution

Given information (part a)

Given that$P\left(C\right)=0.4,P\left(D\right)=0.5andP(C\mid D)=0.6$.

Explanation (part a)

$P\left(CandD\right)$is calculated as

$P\left(CandD\right)=P\left(C|D\right)P\left(D\right)$

Substituting the values, we get

$$\begin{gathered} P\left(CandD\right)=0.6\times 0.5 \\ P\left(CandD\right)=0.30 \end{gathered}$$

Given information (part b)

Given that$P\left(C\right)=0.4,P\left(D\right)=0.5andP(C\mid D)=0.6$.

Explanation (part b)

We have

$$\begin{gathered} P\left(CandD\right)=0.30 \\ \text{Thus},P\left(CANDD\right)\ne 0 \end{gathered}$$

Therefore, events C and D are not mutually exclusive.

Given information (part c)

Given that$P\left(C\right)=0.4,P\left(D\right)=0.5andP(C\mid D)=0.6$.

Explanation (part c)

We have $P\left(C\right)=0.4$

$P\left(C|D\right)=0.6$

Thus,$P\left(C\right)\ne P(C\mid D)$

The first condition is not satisfied.

Therefore, events CandD are not independent.

Given information (part d)

Given that$P\left(C\right)=0.4,P\left(D\right)=0.5andP(C\mid D)=0.6$

Explanation (part d)

$P\left(CorD\right)$is calculated as

$P(C\text{OR}D)=P\left(C\right)+P\left(D\right)-P(C\text{AND}D)$

Substituting the values, we get

$$\begin{gathered} P(CORD)=0.4+0.5-0.3 \\ P(CORD)=0.9-0.3 \\ P(CORD)=0.6 \end{gathered}$$

Given information (part e)

Given that$P\left(C\right)=0.4,P\left(D\right)=0.5andP(C\mid D)=0.6$.

Explanation (part e)

$P\left(D|C\right)$is calculated as

$$\begin{gathered} P\left(DANDC\right)=P(D\mid C)P\left(C\right) \\ P(D\mid C)=\frac{P\left(DANDC\right)}{P\left(C\right)} \end{gathered}$$

Substituting the values, we get

$$\begin{gathered} P\left(D|C\right)=\frac{0.3}{0.4} \\ P\left(D|C\right)=0.75 \end{gathered}$$