Chapter 5.2, Problem 54E

To determine

Probability of intersection of both events C and D.

Answer to Problem 54E

Probability of intersection,

  $P\left(\text{C}\cap \text{D}\right)=0.30$

Explanation of Solution

Given information:

  $P\left(C\right)=0.6$ ,

And

  $P\left(D\right)=0.45$ ,

And

  $P\left(C\cup D\right)=0.75$ .

Calculations:

For any two events, general addition rule:

  $P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right)$

Apply the above general addition rule:

  $P\left(C\cup D\right)=P\left(C\text{or}D\right)=P\left(C\right)+P\left(D\right)-P\left(C\text{and}D\right)=P\left(C\right)+P\left(D\right)-P\left(\text{C}\cap \text{D}\right)$

Thus,

The equation we obtain

  $P\left(C\cup D\right)=P\left(C\right)+P\left(D\right)-P\left(\text{C}\cap \text{D}\right)$

Add $P\left(\text{C}\cap \text{D}\right)$ to both sides:

  $P\left(C\cup D\right)+P\left(C\cap D\right)=P\left(C\right)+P\left(D\right)$

Subtract $P\left(C\cup D\right)$ from both sides:

  $P\left(C\cap D\right)=P\left(C\right)+P\left(D\right)-P\left(C\cup D\right)$

Evaluate the above equation by substituting the values:

  $\begin{array}{l}P\left(C\cap D\right)=P\left(C\right)+P\left(D\right)-P\left(C\cup D\right)\\ =0.6+0.45-0.75\\ =0.30\end{array}$