Question

From the deck of 52 cards, face clubs are removed. Then the remaining cards are well shuffled and a card is drawn at random. Find the probability that the drawn card is a heart card.
(a) \[\dfrac{1}{4}\]
(b) \[\dfrac{13}{49}\]
(c) \[\dfrac{3}{52}\]
(d) \[\dfrac{49}{52}\]

Answer 1

Hint: We solve this problem by using the cards that are removed and the remaining cards. In a deck of 52 cards there will be 4 sets of 13 cards each named clubs, spades, hearts and diamonds in which each set has 3 face cards, one ace and numbered cards from 2 to 10.
By using this information first we find the remaining cards after removing face club cards. Then we can find the probability of getting a heart by using the formula of probability that is
\[\Rightarrow P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]

Complete step-by-step answer:
We are given that the face club cards are removed from the deck of 52 cards.
We know that in a deck of 52 cards there will be 4 sets of 13 cards each named clubs, spades, hearts and diamonds in which each set has 3 face cards, one ace and numbered cards from 2 to 10.
So, from this information we can say that a total of 3 cards are removed from the deck of 52 cards because there will be only 3 face cards in clubs.
So, let us calculate the number of remaining cards after removing 3 cards from 52 cards then we get the number of remaining cards as 49.
Now, we are given that a card is drawn at random from the remaining cards.
We are asked to find the probability that the drawn card is a heart card.
Let us assume that the number of possible outcomes as\['n'\]
We know that there will be 13 cards in the set of hearts.
Here, we can see that we removed club cards only. So, we get the possible outcomes as
\[\Rightarrow n=13\]
Let us assume that the total number of outcomes as \['N'\]
Here, we can see that there are total of 49 remaining cards so, we get the total number of possible outcomes as
\[\Rightarrow N=49\]
Let us assume that the probability of getting the heart card as \['P'\]
We know that the formula of probability that is
\[\Rightarrow P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By using the above formula we get the required probability as
\[\Rightarrow P=\dfrac{n}{N}\]
Now, by substituting the required values in the above equation we get
\[\Rightarrow P=\dfrac{13}{49}\]
Therefore, we can conclude that the required probability as \[\dfrac{13}{49}\]

So, the correct answer is “Option B”.

Note: Students may make mistakes in taking the total number of possible outcomes.
We are given that the card is drawn at random from the remaining cards. So, we get the total number of possible outcomes as
\[\Rightarrow N=49\]
But students may do mistake and take the total number of possible outcomes as
\[\Rightarrow N=52\]
This is because they assume that the card drawn is from the original deck. But we need to draw the card from the remaining cards.
This part needs to be taken care of.