Chapter 15.10, Problem 14WE

(a)

To determine

To find: The independent events $A$ and $B$ .

Answer to Problem 14WE

The two events $A$ and $B$ 9 are independent Drawing a red card is independent of drawing an ace.

Explanation of Solution

Given information:

The given information isa card is drawn from a $52$ -card deck. Let $A$ be the event that the card is red, $B$ be the event that it is an ace, and $C$ be the event that it is a $9$ or $10$ .

Calculation:

We find whether the events $A$ and $B$ are independent.

Two events $A$ and $B$ Are independent if and only if,

  $P\left(A\cap B\right)=P\left(A\right).P\left(B\right)$

First we find the probabilities of the events $A$ , $B$ and $A\cap B$ .

There are $26$ red cards in a $52$ -card deck. Hence the event A consists of $26$ elements.

  $\begin{array}{l}P\left(A\right)=\frac{26}{52}\\ P\left(A\right)=\frac{1}{2}\end{array}$

There are $4$ aces in a $52$ -card deck. Hence the event B consists of $4$ elements

  $\begin{array}{l}P\left(B\right)=\frac{4}{52}\\ P\left(B\right)=\frac{1}{13}\end{array}$

The event $P\left(A\cap B\right)$ consists of the aces which are red. There are $2$ red aces (and $2$ black aces).

  $\begin{array}{l}P\left(A\cap B\right)=\frac{2}{52}\\ P\left(A\cap B\right)=\frac{1}{26}\end{array}$

  $\begin{array}{l}P\left(A\right).P\left(B\right)=\frac{1}{2}\times \frac{1}{13}\\ P\left(A\right).P\left(B\right)=\frac{1}{26}\end{array}$

Thus we have $P\left(A\cap B\right)=P\left(A\right).P\left(B\right)$ .

Therefore, the two events $A$ and $B$ are independent. Drawing a red card is independent of drawing an ace.

(b)

To determine

To find:The independent events $A$ and $C$ .

Answer to Problem 14WE

The two events $A$ and $C$ are independent.

Explanation of Solution

Given information:

The given information isa card is drawn from a $52$ -card deck. Let $A$ be the event that the card is red, $B$ be the event that it is an ace, and $C$ be the event that it is a $9$ or $10$ .

Calculation:

We find whether the events $A$ and $B$ are independent.

Two events $A$ and $B$ Are independent if and only if,

  $P\left(A\cap B\right)=P\left(A\right).P\left(B\right)$

First we find the probabilities of the events $A$ , $C$ and $A\cap C$ .

There are $26$ red cards in a $52$ -card deck. Hence the event A consists of $26$ elements.

  $\begin{array}{l}P\left(A\right)=\frac{26}{52}\\ P\left(A\right)=\frac{1}{2}\end{array}$

There are four 9s and four 10s in a $52$ -card deck. Hence the event C consists of 8 elements.

  $\begin{array}{l}P\left(C\right)=\frac{8}{52}\\ P\left(C\right)=\frac{2}{13}\end{array}$

The event $P\left(A\cap C\right)$ consists of the 9s or 10s which are red. There are two 9s which are red (and 2 Black 9s) and two 10s which are red and 2 black 10s).

  $\begin{array}{l}P\left(A\cap C\right)=\frac{4}{52}\\ P\left(A\cap C\right)=\frac{1}{13}\end{array}$

  $\begin{array}{l}P\left(A\right).P\left(C\right)=\frac{1}{2}\times \frac{2}{13}\\ P\left(A\right).P\left(C\right)=\frac{1}{13}\end{array}$

Thus we have $P\left(A\cap C\right)=P\left(A\right).P\left(C\right)$ .

Therefore, the two events $A$ and $C$ are independent Drawing a red card is independent of drawing a 9 or 10.

(c)

To determine

To find: The independent events $B$ and $C$ .

Answer to Problem 14WE

The two events $B$ and $C$ are independent

Explanation of Solution

Given information:

The given information isa card is drawn from a $52$ -card deck. Let $A$ be the event that the card is red, $B$ be the event that it is an ace, and $C$ be the event that it is a $9$ or $10$ .

Calculation:

We find whether the events $A$ and $B$ are independent.

Two events $A$ and $B$ Are independent if and only if,

  $P\left(A\cap B\right)=P\left(A\right).P\left(B\right)$

First we find the probabilities of the events $B$ , $C$ and $B\cap C$ .

There are $4$ aces in a $52$ -card deck. Hence the event consists of $4$ elements.

  $\begin{array}{l}P\left(B\right)=\frac{4}{52}\\ P\left(B\right)=\frac{1}{13}\end{array}$

There are four 9s and four 10s in a $52$ -card deck. Hence the event C consists of 8 elements.

  $\begin{array}{l}P\left(C\right)=\frac{8}{52}\\ P\left(C\right)=\frac{2}{13}\end{array}$

There are no outcomes which are common to the events $B$ and $C$ .Thus $B\cap C$ an empty set.

Hence

  $\begin{array}{l}P\left(B\cap C\right)=P\left(\phi \right)\\ P\left(B\cap C\right)=0\end{array}$

  $\begin{array}{l}P\left(B\right).P\left(C\right)=\frac{1}{13}\times \frac{2}{13}\\ P\left(A\right).P\left(C\right)=\frac{2}{169}\end{array}$

Thus we have $P\left(B\cap C\right)=P\left(B\right).P\left(C\right)$ .

Therefore, the two events $B$ and $C$ are independent Drawing a red card is independent of drawing a 9 or 10.