Probability Question

Question: A standard deck of cards has 52 cards, divided into 4 suits (hearts, diamonds, clubs, spades) with 13 cards each. If you draw one card at random from the deck, what is the probability that the card drawn is:

A heart?
A queen?
A heart or a queen?
A queen of hearts?

Solutions

1. Probability of Drawing a Heart

There are 13 hearts in a deck of 52 cards.

$cards=1352=14=0.25P(\text{Heart}) = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4} = 0.25$

2. Probability of Drawing a Queen

There are 4 queens in a deck of 52 cards (one queen per suit).

$cards=452=113≈0.077P(\text{Queen}) = \frac{\text{Number of queens}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \approx 0.077$

3. Probability of Drawing a Heart or a Queen

We need to use the formula for the union of two events: $\cup B) = P(A) + P(B) – P(A \cap B)$ .

$P(Heart)=1352=0.25P(\text{Heart}) = \frac{13}{52} = 0.25$
$P(Queen)=452≈0.077P(\text{Queen}) = \frac{4}{52} \approx 0.077$
$Hearts)=152≈0.019P(\text{Heart and Queen}) = P(\text{Queen of Hearts}) = \frac{1}{52} \approx 0.019$

$Queen)P(\text{Heart or Queen}) = P(\text{Heart}) + P(\text{Queen}) – P(\text{Heart and Queen})$

$Queen)=0.25+0.077−0.019=0.308P(\text{Heart or Queen}) = 0.25 + 0.077 – 0.019 = 0.308$

So, the probability of drawing a heart or a queen is approximately $0.308$ (or $30.8%30.8\%$ ).

4. Probability of Drawing the Queen of Hearts

There is only one queen of hearts in the deck.

$Hearts)=152≈0.019P(\text{Queen of Hearts}) = \frac{1}{52} \approx 0.019$

So, the probability of drawing the queen of hearts is approximately $0.019$ (or $1.9%1.9\%$ ).

Summary

The probability of drawing a heart is $0.25$ (or $25%25\%$ ).
The probability of drawing a queen is $0.077$ (or $7.7%7.7\%$ ).
The probability of drawing a heart or a queen is $0.308$ (or $30.8%30.8\%$ ).
The probability of drawing the queen of hearts is $0.019$ (or $1.9%1.9\%$ ).