A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = 0.40, P(D) = 0.30 and P(D|B) = 0.5.

a. Find P(B′).

b. Find P(D AND B).

c. Find P(B|D).

d. Find P(D AND B′).

e. Find P(D|B′).

A student goes to the library. Let events $B=$the student checks out a book and D= the student checks out a DVD. Suppose that $P\left(B\right)=0.40,P\left(D\right)=0.30$and $P(D\mid B)=0.50$.

We have

$\begin{array}{r}P\left(B\right)=0.40\\ P\left(D\right)=0.30\\ P(D\mid B)=0.50\end{array}$

We need to calculate $P\left(B^{\text{'}}\right)$

$P\left(B^{\text{'}}\right)=1-P\left(B\right)$

Substituting the values, we get

$P\left(B^{\text{'}}\right)=1-0.40$

$=0.60$

We have

$P\left(B\right)=0.40$

$P(D\mid B)=0.50$

We need to calculate $P(DANDB)$

$P(\mathrm{D}ANDB)=P(D\mid B)\times P\left(B\right)$

Substituting the values, we get

$\begin{array}{r}P(\mathrm{D}ANDB)=0.5\times 0.4\\ P(DANDB)=0.2\end{array}$

We have

$\begin{array}{r}P\left(B\right)=0.40\\ P\left(D\right)=0.30\\ P(DANDB)=0.2\\ P(D\mid B)=0.50\end{array}$

We need to calculate $P(B\mid D)$

role="math" localid="1648032689213" $P(B\mid D)=\frac{P(DANDB)}{P\left(D\right)}$

Substituting the values, we get

$\begin{array}{r}P(B\mid D)=\frac{0.2}{0.30}\\ P(B\mid D)=0.67\end{array}$

We have

$\begin{array}{r}P\left(B\right)=0.40\\ P\left(D\right)=0.30\\ P(D\mid B)=0.50\end{array}$

We need to calculate $P\left(DAND{B}^{\text{'}}\right)$

$P\left(DAND{B}^{\text{'}}\right)=P\left(D\right)-P(DANDB)$

Substituting the values, we get

$P\left(DAND{B}^{\text{'}}\right)=0.3-0.2$

$\begin{array}{r}P\left(DAND{B}^{\mathrm{\prime }}\right)=0.3-0.2\\ P\left(DAND{B}^{\mathrm{\prime }}\right)=0.1\end{array}$

We have

$\begin{array}{r}P\left(B\right)=0.40\\ P\left(D\right)=0.30\\ P\left(DANDB\right)=0.2\\ P(D\mid B)=0.50\end{array}$

We need to calculate $P\left(D\mid B^{\text{'}}\right)$

$P(B\mid D)=\frac{P(DANDB)}{P\left(D\right)}$

Substituting the values, we get

$\begin{array}{r}P(B\mid D)=\frac{0.2}{0.30}\\ P\left(D\mid B^{\mathrm{\prime }}\right)=0.06\end{array}$