Question

A card is drawn from a well shuffled pack of 52 cards. Find the probability that the card drawn is
(a) a red face card.
(b) neither a club nor a spade.
(c) neither an ace nor a king of red colour.

Answer 1

Hint- In this question, first we will see the definition of probability. A deck of cards has 52 cards in total. So, for each of the given statements, we will calculate the total number of favourable events and then calculate the probability for each case.

Complete step by step solution:
Probability of an event indicates how likely an event can occur. Mathematically, we can write:
$P(E)={\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total possible outcomes}}}$
(a) a red face card
A deck of cards consists of 52 cards.
It includes 13 ranks in each of four suits( Diamond, Heart, Spade, Club)
Diamond and heart belong to red cards and spade and club belong to black cards.
Therefore, total number of possible events = 52
There are a total of 12 face cards. Out of which 6 are red and 6 are black.
So, total number of favourable events = 6
Let E be the event of occurring a red face card.
$\therefore$ $P(E)={\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total possible outcomes}}}={\displaystyle \frac{6}{52}}={\displaystyle \frac{3}{26}}$.
(b) neither a club nor a spade
Total number of spade = 13
Total number of club = 13
Total number of cards which are neither spade nor club = 52-(13+13)= 26.
$\therefore$Total number of favourable events = 26
Let E be the event of occurring a card which is neither spade nor club.
$\therefore$ $P(E)={\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total possible outcomes}}}={\displaystyle \frac{26}{52}}={\displaystyle \frac{1}{2}}$
(c) ) neither an ace nor a king of red colour
Total number of ace of red colour = 2
Total number of king of red colour =2
Total number of cards which are neither an ace nor a king of red colour=52-4 = 48.
$\therefore$Total number of favourable events = 48
Let E be the event of occurring a card which is neither an ace nor a king of red colour.
$\therefore$ $P(E)={\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total possible outcomes}}}={\displaystyle \frac{48}{52}}={\displaystyle \frac{24}{26}}$

Note- In such a type of question, the most important thing is to find the total number of favourable events. For this you must have knowledge about a deck of cards. In a deck of cards, there are 13 ranks. Each rank has four further suits(Diamond, Heart, Spade, Club).The probability of an event always lies between 0 and 1.