There are 13 clubs in a standard 52-card deck of playing cards. There are also 4 jacks: 1 jack of spades, 1 jack of hearts, 1 jack of diamonds, and 1 jack of clubs.

What is the probability that a card picked at random from a standard deck of playing cards is a club or a jack?

A. [tex]$\frac{4}{13}$[/tex]

B. [tex]$\frac{7}{28}$[/tex]

C. [tex]$\frac{3}{18}$[/tex]

D. [tex]$\frac{1}{32}$[/tex]



Answer :

To determine the probability that a card picked at random from a standard deck of playing cards is either a club or a jack, we can use the principle of set theory. Here's the step-by-step process:

1. Identify the Total Number of Cards in the Deck:
- A standard deck contains 52 cards.

2. Determine the Number of Clubs in the Deck:
- There are 13 clubs in a standard deck, one for each rank (2 through Ace).

3. Determine the Number of Jacks in the Deck:
- There are 4 jacks in a standard deck (Jack of Spades, Jack of Hearts, Jack of Diamonds, and Jack of Clubs).

4. Identify the Overlap Between Clubs and Jacks:
- One of the jacks is also a club, specifically the Jack of Clubs. This card has been counted both among the 13 clubs and the 4 jacks, so we must ensure we don't double-count it.

5. Calculate the Number of Cards that are Either Clubs or Jacks:
- Add the number of clubs and the number of jacks, then subtract the one card that is both a club and a jack to avoid double-counting:

Number of clubs = 13

Number of jacks = 4

Overlap (Jack of Clubs) = 1

So, the number of favorable outcomes (either a club or a jack) is:
[tex]\[ \text{Number of clubs} + \text{Number of jacks} - \text{Overlap} = 13 + 4 - 1 = 16 \][/tex]

6. Determine the Total Number of Possible Outcomes:
- There are 52 cards in total.

7. Calculate the Probability:
- The probability of drawing a card that is either a club or a jack is the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{16}{52} \][/tex]

8. Simplify the Fraction:
- To simplify [tex]\(\frac{16}{52}\)[/tex], divide both the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[ \frac{16 \div 4}{52 \div 4} = \frac{4}{13} \][/tex]

Therefore, the probability that a card picked at random from a standard deck of playing cards is a club or a jack is [tex]\(\frac{4}{13}\)[/tex].

So, the correct answer is:
A. [tex]\(\frac{4}{13}\)[/tex]