Question

: From a well shuffled pack of $52$ cards, one card is drawn at random. Find the probability of getting a diamond.

Answer 1

Hint : There are total $52$ cards in a deck with $4$ different suits: ace, diamonds, hearts and clubs. All the suits have equal number of cards and each suit has $13$ cards. Hence to arrive at the probability, we will have to divide $13$ by $52$ with the formula $P(E) = \dfrac{{Number\,of\,favourable\,outcomes}}{{Total\,number\,of\,outcomes}}$ .

Complete step-by-step answer :

We are given the figure as follows, represented as a well shuffled pack of $52$ cards, one card is drawn at random. Find the probability of getting a diamond.
We will use combination techniques to find out the solution. A combination is a mathematical technique for calculating the number of possible arrangements in a collection of items where the order of the items is irrelevant. It is denoted as $n{C_r}$ where $n$ is the total number of objects and $r$ is the number of selections.
The formula to find combination is given as follows:
$n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Moreover, probability of an event can be obtained as ratio of occurrence of an event to the total sample size. The formula of finding probability of a favourable event is $P(E) = \dfrac{{Number\,of\,favourable\,outcomes}}{{Total\,number\,of\,outcomes}}$
Now we can proceed to solve the sum as follows:
Total cards in pack are $52$ . Hence the total combination of selecting one card will be-
$n{C_r} = \dfrac{{52!}}{{1!(52 - 1)!}}$
$= \dfrac{{52!}}{{1!(51)!}}$
$= \dfrac{{52 \times 51!}}{{1!(51)!}}$
$= 51{C_1}$
Total number of diamonds in a pack are $13$ starting from A to $10$ , King, Queen and Jack. Hence the total combination of selecting one card of diamond will be-
$n{C_r} = \dfrac{{13!}}{{1!(13 - 1)!}}$
$= \dfrac{{13!}}{{1!(12)!}}$
$= \dfrac{{13 \times 12!}}{{1!(12)!}}$
$= 13{C_1}$
Now let the probability of getting a diamond be $P(D)$ :
$P(D) = \dfrac{{Number\,of\,getting\,diamond}}{{Total\,number\,of\,cards}}$
$= \dfrac{{13{C_1}}}{{52{C_1}}}$
$= \dfrac{{13}}{{52}}$
$= \dfrac{1}{4}$
In terms of percentage, it will be $\dfrac{1}{4} \times 100 = 25\%$
So, the correct answer is “ $= \dfrac{1}{4}$ ”.

Note : We can solve the sum directly without applying the combination formula since there is selection of only one card.
We know that there are $13$ diamond cards so favourable outcomes will be $13$ and sample size will be $52$ as the total number of cards. We use the probability formula to obtain the required solution.