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
`((1)/(221))`

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}} = (4)/(52) = ((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}} = (3)/(51)`

⇒ The combined probability of drawing 2 kings simultaneously

=
`((1)/(13) )* ((3)/(51) ) = (3)/(663) = ((1)/(221))`

This is true for all same ranked cards.

Hence, the probability that Ben draws 2 cards of the same rank is
`((1)/(221))`