# 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. - Mathematics

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.

#### Solution

Out of 52 cards, four cards can be randomly chosen in 52C4  ways.
∴ n(S) = 52C4
Let A = event where the four cards drawn are red
and B = event where the four cards drawn are black
Then, n(A) = 26C4 and n(B) = 26C4

$\Rightarrow P\left( A \right) = \frac{^{26}{}{C}_4}{^{52}{}{C}_4}$  and
$P\left( B \right) = \frac{^{26}{}{C}_4}{^{52}{}{C}_4}$
A and B are mutually exclusive events.
i.e. P (A ∩ B) = 0
P (A ∪ B) = P(A) + P (B) - P (A ∩ B)

= $\frac{^{26}{}{C}_4}{^{52}{}{C}_4} + \frac{^{26}{}{C}_4}{6{52}{}{C}_4}$ -  0
=$\frac{46}{17 \times 49} + \frac{46}{17 \times 49}$

= $2 \times \frac{46}{17 \times 49} = \frac{92}{833}$

Hence, the probability that all the drawn cards are of the same colour is $\frac{92}{833}$ .

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 33 Probability
Exercise 33.4 | Q 15 | Page 68