Question

Red kings, queens and jacks are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the card drawn is
(i) a king
(ii) of red colour
(iii) a spade

Answer 1

Hint: Find the total number of cards left in the deck after removing the given cards. Then find the total number of ways in which one card can be selected. Find the number of possible ways in each part and apply the formula of probability, which is $\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of possible outcomes}}}}$

Complete step-by-step answer:
We have total 52 cards in a deck.
We are given that the red kings, queens and jacks are removed from the deck.
Now we know that there are 2 red kings, 2 red queens, and 2 red jacks, where 1 belongs to hearts and 1 is of diamonds.
This means we have removed 6 from cards from the deck of 52 cards and now there are 46 cards left in the deck.
Also, we have to select one card from 46 left cards. Therefore, total number of possible outcomes are $^{46}{C_1}$.
In part (i) we have to find the probability of selecting a king from the remaining cards.
There are 2 Kings left in the deck. Hence the number of ways we can select a king is $^2{C_1}$
As we know that the probability of the any event is given by $\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of possible outcomes}}}}$
Hence the probability of getting a king is \[\dfrac{{^2{C_1}}}{{^{46}{C_1}}}\]
Also, we know that \[^n{C_1} = n\]
Therefore, the probability of getting a king is,
$\dfrac{2}{{46}} = \dfrac{1}{{23}}$
In part (ii) we have to find the probability of selecting a red colour card.
There are 26 red cards in a normal deck. But here we have removed 2 red kings, 2 red queens and 2 red jacks. So there are 22 red cards left in the deck.
We have to select one red card.
Hence the number of ways of selecting a red card is \[^{22}{C_1}\].
Then the probability of getting a red card is \[\dfrac{{^{22}{C_1}}}{{^{46}{C_1}}}\]
Which is equal to $\dfrac{{22}}{{46}} = \dfrac{{11}}{{23}}$

In part (iii) we have to find the probability of selecting a spade.
We have not removed any spade card, and there are 13 spade cards in the deck.
Number of ways to select one spade card is $^{13}{C_1}$.
Therefore the probability is of selecting a spade is \[\dfrac{{^{13}{C_1}}}{{^{46}{C_1}}}\]
Hence, the probability of getting a spade is $\dfrac{{13}}{{46}}$.

Note: One must know that there are 52 cards in the deck, out of which 26 are red and 26 are black.
There are two red suites, diamond and hearts each having 13 cards. Similarly, there are two suites of black cards, spade and club having 13 cards each. Each suit has 3 face cards and thus there are a total 12 face cards in the whole deck.