Question

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.

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

Short answer

Yes, the two events are independent.

Step-by-step solution

Step 1. Given information

The table is:

Step 2. Concept used

The condition for the independent events is:

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

Step 3. Calculation

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

Here,

$$\begin{gathered} P\left(X\right)=\frac{Numberofheartcards}{Totalcards} \\ =\frac{13}{52} \\ =\frac{1}{4} \end{gathered}$$

$$\begin{gathered} P\left(Y\right)=\frac{Numberoffacecards}{Totalcards} \\ =\frac{12}{52} \\ =\frac{3}{13} \\ P(X\cap Y)=\frac{3}{32} \end{gathered}$$

Now check the required condition as:

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

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