Question

Ben randomly draws 2 cards without replacement, from a standard deck of playing cards, What is the probability that ben draws 2 cards of the same rank? Express your answer as a common fraction

Answer 1

The probability that Ben draws 2 cards of the same rank is $(\frac{1}{221})$

Step-by-step explanation:

Here, the total number of cards in a deck = 52

Now, let us assume the 2 same ranked cards drawn are Kings.

So, P( drawing a first king)  = $\frac{\textrm{Total number of kings in the deck}}{\textrm{Total number of cards}} = \frac{4}{52} = (\frac{1}{13} )$

Now, the drawn card is not replaced.

So, P( drawing a second king)  = $\frac{\textrm{Total number of kings left}}{\textrm{Total number of cards}} = \frac{3}{51}$

⇒ The combined probability of drawing 2 kings simultaneously

   = $(\frac{1}{13} )\times (\frac{3}{51} ) = \frac{3}{663} = (\frac{1}{221})$

This is true for all same ranked cards.

Hence, the probability that Ben draws 2 cards of the same rank is $(\frac{1}{221})$