Chapter 5.3, Problem 72E

To determine

To check: Whether the events “face card” and “heart” are independent.

Answer to Problem 72E

Yes, the two events are independent.

Explanation of Solution

Given:

The table is:

	Type of card
	Face card	Non face card	Total
Heart	3	10	13
Non heart	9	30	39
Total	12	40	52

Concept used:

The condition for the independent events is:

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

Calculation:

Consider, X be the event that shows the heart card and Y be the event that shows the face card.

Here,

  $\begin{array}{c}P\left(X\right)=\frac{\text{Number of heart cards}}{\text{Total cards}}\\ =\frac{13}{52}\\ =\frac{1}{4}\end{array}$

  $\begin{array}{c}P\left(Y\right)=\frac{\text{Number of face cards}}{\text{Total cards}}\\ =\frac{12}{52}\\ =\frac{3}{13}\end{array}$

  $P\left(X\cap Y\right)=\frac{3}{52}$

Now check the required condition as:

‘ $\begin{array}{c}P\left(X\cap Y\right)=P\left(X\right)\times P\left(Y\right)\\ \frac{3}{52}=\frac{1}{4}\times \frac{3}{13}\\ \frac{3}{52}=\frac{3}{52}\end{array}$

Since, the required condition is satisfied. Thus, it could be said that the events “face card” and “heart” are independent.