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If $2$ cards are drawn from a well shuffled pack of $52$ cards, the probability that they are of the same colours without replacement, is

\[\begin{array}{1 1}(1)\frac{1}{2}&(2)\frac{26}{51}\\(3)\frac{25}{51}&(4)\frac{25}{102}\end{array}\]

1 Answer

Total number of cards $=52$
Number of ways choosing 2 cards from $52\; cards = 52\;C_2$
26 Black chords +26 Red chords
Number of ways of choosing 2 black chords out of 26 Black chords $=26\;C_2$
Number of ways of choosing 2 Red chords from 26 red chords $=26\;C_2$
Probability getting 2 chords of same colour =Probability getting 2 black chords 26 Black chords.
(or) Probability getting 2 Red chords from 26 Red chords .
$\qquad= \large\frac{26 C_2 +26 C_2}{(52 C_2)}$
$\qquad= \large\frac{\Large\frac{26 \times 25}{1 \times 2} + \frac{26 \times 25}{1 \times 2}}{\bigg( \Large\frac{52 \times 5I}{1 \times 2} \bigg)}$
$\qquad= \large\frac{26 \times 25 +26 \times 25}{52 \times 51}$
$\qquad= \large\frac{ 2 \times 26 \times 25}{52 \times 51}$
$\qquad= \large\frac{25}{51}$
Hence 3 is the correct answer.
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