Question

Yand Zare independent events.

a. Rewrite the basic Addition Rule $P(YORZ)=P\left(Y\right)+P\left(Z\right)-P(YANDZ)$using the information that Y and Z are independent events.

b. Use the rewritten rule to findP(Z) if $P(YORZ)=0.71$and $P\left(Y\right)=0.42$

Short answer

(a) $P(YORZ)=P\left(Y\right)+P\left(Z\right)-P\left(Y\right)P\left(Z\right)$.

(b)$P\left(Z\right)=0.5$.

Step-by-step solution

Given information (part a)

Y andZ are independent events.

Explanation (part a)

The addition rule states that

$P\left(YorZ\right)=P\left(Y\right)+P\left(Z\right)-P\left(YandZ\right)$

Also, given that Yand Z are independent events

Thus, we have $P\left(YandZ\right)=P\left(Y\right)P\left(Z\right)$

Substituting the value in the above rule, we get

$P\left(YorZ\right)=P\left(Y\right)+P\left(Z\right)-P\left(Y\right)P\left(Z\right)$

Given information (part b)

Y and Z are independent events.$P\left(YorZ\right)=0.71\text{and}P\left(Y\right)=0.42$

Explanation (part b)

The addition rule states that

$P(YORZ)=P\left(Y\right)+P\left(Z\right)-P(YANDZ)$

Also, given that Y andZ are independent events

Thus, we have $P\left(YandZ\right)=P\left(Y\right)P\left(Z\right)$

Substituting the value in the above rule, we get

$$\begin{gathered} P(Y\cap Z)=P\left(Y\right)+P\left(Z\right)-P\left(Y\right)P\left(Z\right)\mathrm{P}(\mathrm{Y}\cap Z) \\ =\mathrm{P}\left(\mathrm{Y}\right)+\mathrm{P}\left(\mathrm{Z}\right)-\mathrm{P}\left(\mathrm{Y}\right)\mathrm{P}\left(\mathrm{Z}\right) \end{gathered}$$

Now, we have

$$\begin{gathered} P\left(YorZ\right)=0.71 \\ P\left(Y\right)=0.42 \end{gathered}$$

Substituting the values, we get

$$\begin{gathered} 0.71=0.42+P\left(Z\right)-(0.42\times P(Z\left)\right) \\ 0.71=0.42+\left[P\right(Z\left)\right(1-0.42\left)\right] \\ 0.71=0.42+0.58P\left(Z\right) \\ P\left(Z\right)=\frac{0.71-0.42}{0.58} \\ P\left(Z\right)=0.5 \end{gathered}$$