Question

From a well shuffled deck of $52$ cards, $4$ cards are drawn at random. What is the probability that all the drawn cards are of the same colour?

Answer 1

Hint : The given question requires us to find the probability of drawing $4$ cards of same colour from normal deck of cards by the combination technique, formula for this technique is $n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ . We can solve the sum by first finding out the number of ways of choosing $4$ cards and then dividing it by the number of ways of choosing any card from the deck to arrive at the solution.

Complete step-by-step answer :
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.
We can proceed to solve the sum as follows:
We know that there are total $52$ cards in a deck having $4$ suits: diamond, space, hearts and club. Each suit will have $\dfrac{{52}}{4} = 13$ cards.
Total number of ways of selecting any four cards from the deck will be represented as $52{C_4}$ (It can also be written as $^{52}{C_4}$ ). According to formula it will be as follows:
$n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{52!}}{{4!(52 - 4)!}} = \dfrac{{52!}}{{4!48!}} = \dfrac{{52 \times 51 \times 50 \times 49 \times 48!}}{{4!48!}} = 52{C_4}$
Now we can choose four cards from a single suit of $13$ cards in $13{C_4}$ :
$n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{13!}}{{4!(13 - 4)!}} = \dfrac{{13!}}{{4!9!}} = \dfrac{{13 \times 12 \times 11 \times 10 \times 9!}}{{4!9!}} = 13{C_4}$
Therefore, we can select four cards from $4$ suits of $13$ cards in $4(13{C_4})$ ways.
We can conclude that our sample size is $52{C_4}$ and the number of ways of selecting the cards is $4(13{C_4})$ . To find the probability, we will find their ratio as follows:
Hence the probability of selecting four cards of same colour can be calculated as follows:
$= \dfrac{{4(13{C_4})}}{{52{C_4}}}$
$= \dfrac{{4(\dfrac{{13 \times 12 \times 11 \times 10}}{{4 \times 3 \times 2 \times 1}})}}{{(\dfrac{{52 \times 51 \times 50 \times 49}}{{4 \times 3 \times 2 \times 1}})}}$
$= \dfrac{{92}}{{833}}$
Hence the probability of selecting $4$ cards of same colour from deck of $52$ cards is $\dfrac{{92}}{{833}}$ or $\dfrac{{92}}{{833}} \times 100 = 11.04\%$
So, the correct answer is $\dfrac{{92}}{{833}}$ ”.

Note : There are certain assumptions made while solving this problem which should be taken care of like:
The cards cannot be replaced in the deck after selecting the card.
There is no bias in selection i.e. the back of the cards are identical and well shuffled before selecting.
There is no extra card other than four suits i.e. there is no joker.