# Probability and playing cards

In this chapter, we will discuss the calculation of probability in the experiment of drawing playing cards.

At the end, some solved examples are given for better understanding of concept.

Let us first discuss the basic concept first.

## Playing cards probability

The playing cards consists of total 52 cards which is divided into four groups of 13 card each.

Given below is the image of all the groups.

As mentioned above, all the group contain 13 cards each. Given below is the description of cards for each group.

Note that there are total of 12 face cards (Jack, Queen and King), three on each group. Given below is the image of face cards for reference.

Here, K = King, J = Jack and Q = Queen.
These cards have illustrations of king, jack and queen, that’s why they are called face cards.

I hope you understood the basic concept of playing cards. Let us now solve some probability questions related to the concept.

### Problems on Probability and Playing cards

Question 01
From the pack of 52 cards, one card is selected at random. Find the probability of getting;

(a) an ace
(b) red card
(c) either red card or King
(d) red card and king

Solution
Total number of outcomes = 52

(a) getting an ace card

We know that there are 4 ace card on entire deck.

Number of favorable outcome = 4

Probability = Favorable outcome / Total outcome

Probability = 4 / 52 = 1/13

Hence, 1/13 is the required probability.

(b) getting a red card

Red diamond card = 13
Red heart card = 13

Total red card = 13 + 13 = 26

Probability = favorable outcome / total outcome

Probability = 26 / 52 = 1/2

Hence, 1/2 is the required probability.

(c) getting either red or king

Total red cards = 26
Total king card = 4

Probability for red card = 26 / 52 (calculated above)

Probability for king card = 4 / 52

Note that while calculating both the probability, the two “red king card” is common in both calculation.

Probability (Red ∩ King card) = 2 / 52

Since there are 2 cards common in above probability, they are mutually non exclusive event.

Applying the formula;

\mathtt{P\ ( A\ \cup \ B\ ) \ =\ P( A) \ +\ P( B) -P( A\cap B)}\\\ \\ \mathtt{P\ ( A\ \cup \ B\ ) \ =\ \frac{26}{52} +\frac{4}{52} -\frac{2}{52}}\\\ \\ \mathtt{P\ ( A\ \cup \ B\ ) \ =\ \frac{28}{52} =\frac{7}{13}} \\ \\

Hence, 7/13 is the required solution.

(d) getting red card and king

Number of favorable outcome = 2

Probability (red card & king) = 2/52 = 1/26

Hence, 1/26 is the required probability.

Question 02
From the pack of 52 cards, a card is drawn randomly. Calculate the probability that its neither a king or Jack ?

Solution
Total number of outcomes = 52

Total number of Kings = 4
Total number of Jacks = 4

Number of favorable outcome = 52 – 4 – 4 = 44

Probability of neither Jack or King = 44 / 52 = 11/13

Question 03
A card is randomly selected from the deck of 52 cards. Find the probability of getting;

(a) Face card
(b) either face card or Diamond
(c) Face card and heart

Solution
Total number of outcomes = 52

(a) Getting face cards

The Jack, Queen and King constituent the face cards. So there are total of 12 face cards in the deck of 52 cards.

Probability of face card = 12 / 52

(b) getting either face cards or diamond

Probability for face cards = 12 / 52

Probability for getting diamonds = 13 / 52

We know that in diamond group, there are three face cards. So the above two events are not mutually exclusive.

P (Face card ∩ diamonds ) = 3 / 52

Now applying mutually exclusive formula;

\mathtt{P\ ( A\ \cup \ B\ ) \ =\ P( A) \ +\ P( B) -P( A\cap B)}\\\ \\ \mathtt{P\ ( A\ \cup \ B\ ) \ =\ \frac{12}{52} +\frac{13}{52} -\frac{3}{52}}\\\ \\ \mathtt{P\ ( A\ \cup \ B\ ) \ =\ \frac{22}{52}}

Hence, 22/52 is the probability of getting either face or diamond card.

(c) probability of getting Face card and heart

There are three face cards in Heart group of cards.

Probability = 3 / 52

Hence, 3/52 is the solution.