Chapter 15, Problem 7E

(a)

To determine

To Calculate: the conditional probability.

Answer to Problem 7E

0.5

Explanation of Solution

Given:

The card is a heart, given that it is red.

Formula used:

  $P\left(H\backslash R\right)=\frac{P\left(H\cap R\right)}{P\left(R\right)}$

Calculation:

Suppose H be the event of drawing heart

And R is the event of drawing red card.

  $P\left(H\backslash R\right)=\frac{P\left(H\cap R\right)}{P\left(R\right)}$

  $\begin{array}{c}P\left(H\cap R\right)=\frac{13}{52}\\ P\left(R\right)=\frac{26}{52}\end{array}$

Putting the values

  $\begin{array}{c}P\left(H\backslash R\right)=\frac{\frac{13}{52}}{\frac{26}{52}}\\ =\frac{13}{26}\\ =0.5\end{array}$

There is 50% possibility of drawing heart given it is red.

(b)

To determine

To Calculate: the conditional probability.

Answer to Problem 7E

100%

Explanation of Solution

Given:

The card is red, given that it is a heart.

Formula used:

  $P\left(H\backslash R\right)=\frac{P\left(H\cap R\right)}{P\left(H\right)}$

Calculation:

Suppose H be the event of drawing heart

Suppose R be the event of drawing red card.

  $P\left(H\backslash R\right)=\frac{P\left(H\cap R\right)}{P\left(H\right)}$

  $\begin{array}{c}P\left(H\cap R\right)=\frac{13}{52}\\ P\left(H\right)=\frac{13}{52}\end{array}$

Putting the values

  $\begin{array}{c}P\left(H\backslash R\right)=\frac{\frac{13}{52}}{\frac{13}{52}}\\ =1\end{array}$

There is 100% possibility of drawing res given it is heart.

(c)

To determine

To Calculate: the conditional probability.

Answer to Problem 7E

0.077

Explanation of Solution

Given:

The card is an ace, given that it is red.

Formula used:

  $P\left(A\backslash R\right)=\frac{P\left(A\cap R\right)}{P\left(R\right)}$

Calculation:

Suppose A be the event of drawing Ace and

R is the event of drawing red.

There are 26 red cards.

There are 4 Aces.

  $\begin{array}{c}P\left(A\cap R\right)=\frac{2}{52}\\ P\left(R\right)=\frac{26}{52}\end{array}$

Putting the values

  $\begin{array}{c}P\left(A\backslash R\right)=\frac{\frac{2}{52}}{\frac{26}{52}}\\ =\frac{2}{26}\\ =0.077\end{array}$

There is about 8% possibility of drawing Ace given that is red

(d)

To determine

To Calculate: the conditional probability.

Answer to Problem 7E

0.33

Explanation of Solution

Given:

The card is a queen, given that it is a face card.

Formula used:

  $P\left(Q\backslash F\right)=\frac{P\left(Q\cap F\right)}{P\left(F\right)}$

Calculation:

Suppose Q be the event of drawing queen and

F is the event of drawing face card.

There are 4 queens and there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings)

Probability that card is queen given that faces that face card is drawn.

  $\begin{array}{c}P\left(Q\backslash F\right)=\frac{P\left(Q\cap F\right)}{P\left(F\right)}\\ P\left(Q\cap F\right)=\frac{4}{52}\\ P\left(F\right)=\frac{12}{52}\end{array}$

Putting the value

  $\begin{array}{c}P\left(Q\backslash F\right)=\frac{\frac{4}{52}}{\frac{12}{52}}\\ =\frac{1}{3}\\ =0.33\end{array}$

There is 33% possibility that queen is drawn given that face card is drawn.