Question

Is this your card? A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits—clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jacks, queens, and kings are referred to as “face cards.” Imagine that we shuffle the deck thoroughly and deal one card. The two-way table summarizes the sample space for this chance process based on whether or not the card is a face card and whether or not the card is a heart.

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

Are the events “heart” and “face card” independent? Justify your answer.

Short answer

Events “heart” and “face card” independent

Step-by-step solution

Given Information

We are given information of cards and a two-way table summarizes the sample space for this chance process based on whether or not the card is a face card and whether or not the card is a heart.

We need to find out are the events “heart” and “face card” independent .

Explanation

Two events are independent if probability of one event does not affect probability of another event.

	Face Cards Non-Face Cards		Total
Heart	$3$	$10$	$13$
Non-Heart	$9$	$30$	$39$
Total	$12$	$40$	$52$

Probability of face card in deck of card=$P$(Face card)=role="math" localid="1653922848957" $\frac{No.offavourableoutcomes}{No.ofpossibleoutcomes}=\frac{\cancel{12}}{\cancel{52}}=\frac{3}{13}$

Probability of heart in deck of card= P(heart)=$\frac{No.offavourableoutcome}{no.oftotaloutcome}=\frac{13}{52}$

Probability of face card and heart=P(face card and heart)=$\frac{3}{52}$

According to definition of conditional probability:

P(Face card | Heart)=$\frac{P(Facecardandheart)}{P\left(Heart\right)}=\frac{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$52$}\right.}}{{\displaystyle \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$52$}\right.}}=\frac{3}{13}$

For event to be independent $P(A|B)=P(A)$

From above probabilities calculated by us we see that : P(Face card)= P(Face card | Heart) = $\frac{3}{13}$

Hence, events “heart” and “face card” independent.