Question

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting neither a club nor a black queen.
$(a){\text{ }}\dfrac{{11}}{{26}} \\\ (b){\text{ }}\dfrac{{19}}{{26}} \\\ (c){\text{ }}\dfrac{6}{{26}} \\\ (d){\text{ }}\dfrac{1}{{26}} \\\$

Answer 1

Hint:In this question use the basic concept of probability which is defined as the ratio of the favorable number of outcomes to the total number of outcomes. The favorable number of outcomes will be the total cases of neither a club nor a black queen and the total number of cases will be 52 as the deck has in total 52 cards.

Complete step-by-step answer:
There are 52 cards in a well shuffled deck.
In where there are 4 suits of cards
Spade, heart, diamond and club.
In all these suits each have 13 cards (king, queen, jack, 10 to 1 and ace).
There are two suits of red color (heart and diamond) and another two suits of black color (spade and club).
So there are 13 cards in the suit of the club.
And a total 4 queens (2 black queens and two red queens) in the deck of well shuffled cards (including the queen of club).
So the total number of club cards and a black queens = 13 + 2 -1 = 14 (as in all 13 club cards one black queen is involved so we have to subtract one (1)).
So the remaining cards which have neither a club nor a black queen = 52 – 14 = 38 cards.
Now as we know that the probability is the ratio of the favorable no of outcomes to the total number of outcomes.
Therefore, P = $\dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
Now the favorable number of outcomes = 38
And the total number of outcomes = 52
Therefore, P = $\dfrac{{{\text{38}}}}{{{\text{52}}}} = \dfrac{{19}}{{26}}$
So this is the required probability.
Hence option (B) is the correct answer.

Note:A deck has 52 cards with 4 houses, each house has 13 cards each, and houses are spade, club, diamond, heart. Heart and diamond are red color houses whereas spade and club are black in color so we have 26 red cards and 26 black cards, now there are 3 face cards in each suit and thus 12 face cards in total. This information is very helpful in solving problems of this kind. Just take the favorable case and divide it by 52 to get the required.