Question

A card is drawn from a well shuffled pack of 52 cards. Find the probability that the card drawn is either a heart, a king or a queen
a. $\dfrac{17}{52}$
b. $\dfrac{21}{52}$
c. $\dfrac{19}{52}$
d. $\dfrac{9}{26}$

Answer 1

Hint: We will use the concept that there are a total of 52 cards in a full deck. And the number of hearts is 13, the number of kings is (4 - 1) = 3, number of queens is (4 - 1) = 3. We will use the formula of the probability that is, $\text{Probability=}\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.

Complete step-by-step answer:
It is given in the question that a card is drawn from a well shuffled pack of 52 cards and we have been asked to find the probability that the card drawn is either a heart, a king or a queen.
We know that there are a total of 52 cards in a deck of cards. We also know that there are a total of 13 heats in a deck of cards.
Also, in a deck of cards, there are a total of 4 kings, but here 1 king is already counted in the 13 hearts, so the number of kings will become (4 - 1) = 3.
Similarly, there are a total of 4 queens, but 1 queen is already counted in 13 hearts, so the number of queens will become (4 - 1) = 3.
Now, we know that the formula of probability is given by, $\text{Probability=}\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.
So, here the favourable outcomes = number of hearts + number of kings + number of queens.
So, we get, favourable outcomes = 13 + 3 + 3 = 19
Also, we know that the total number of outcomes is the total number of cards and that is 52.
Therefore, we can write the probability of a card being drawn to be either a heart, a king or a queen as,
\[\text{Probability=}\dfrac{19}{52}\]
So, the probability that the card drawn is either a heart, a king or a queen is $\dfrac{19}{52}$.
Hence, option (c) is the correct answer.

Note: The most common mistake that the students make while solving this question is that they take the number of king and queen as 4 instead of 3, but it is wrong because, as we have also been given the condition that the card drawn can be a heart, and we know that out of 13 hearts, 1 card is king and 1 card is queen. So, already 1 king and 1 queen is included in the 13 hearts, so we have to take the number of kings and queens as (4 - 1) = 3. So, by writing the number of king and queen as 4, the students may get the answer as $\dfrac{21}{52}$, that is option (b), which is wrong.