Question

In a deck of cards, find the probability of getting a red face card.

Answer 1

Find the total number of cards present in a deck or pack of cards. Assume this number as the total number of sample space denoted by n (S). Now, find the definition of a face card and determine the total number of face cards, that is king, queen and jack, which are red in colour. Assume this number as the total number of favourable outcomes denoted by n (E). Apply the formula: - $P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$ to get the answer. Here, P (E) denotes the probability of an event.

Complete step by step answer:
We have been provided with a deck of cards and we have to find the probability of getting a red face card.
Now, we know that a well shuffled pack or deck of card contains a total of 52 cards. So, the total number of outcomes or total number of sample space denoted by n (S) can be given as: -
$\Rightarrow n\left( S \right)=52$
Now, in a deck of playing cards, the term face cards is generally used to describe a card that depicts a person. We know that a pack of 52 cards contains four suits of 13 cards each: spades, hearts, clubs and diamonds. Among these, clubs and spades are black and spades and hearts are red in colour. Each suit contains three face cards, namely king, queen and jack.
So, there will be a total of 12 face cards, 3 from each suit. Therefore, the number of face cards which are red will be 6 in number. So, the total number of favourable outcomes denoted by n (E) can be given as: -
$\Rightarrow n\left( E \right)=6$
Now, we know that the probability of an event denoted by P (E) is the ratio of number of favourable outcomes to the total number of outcomes. Therefore, we have,
$\Rightarrow$ Probability of getting a red face card = P (E)

& \Rightarrow P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)} \\\ & \Rightarrow P\left( E \right)=\dfrac{6}{52} \\\ & \Rightarrow P\left( E \right)=\dfrac{3}{26} \\\ \end{aligned}$$ **Hence, the required probability is $$\dfrac{3}{26}$$.** **Note:** One may note that we must remember the definitions and arrangement of a deck of cards to solve the above question. The probability of an event is always less than or equal to 1 therefore we must take the ratio $$\dfrac{n\left( E \right)}{n\left( S \right)}$$ and not $$\dfrac{n\left( S \right)}{n\left( E \right)}$$. In the above question if we will not remember the colour of each suit and the number of face cards in each suit then we will not be able to solve the question. Remember that ace is not considered as a face card.