you have a deck of 52 standard playing cards you draw two wh

you have a deck of 52 standard playing cards. you draw two. what is the chance that at least one is a heart and at least one is a numbered card? Ace is considered to be a numbered card as well.

Solution

Since two cards are drawn at random, the total sample space is ⁵²C₂ ways.

Atleast one is a heart: there are 13 heart cards in a full pack, so one out of 13 would be ¹³C₁ ways.

Atleast one is a number card in which ace is also included: As there are 9 numbered cards from 2 to 9, including ace will add up to 10 numbered cards from each parts of heart, spade, club and diamond.

So it addes up to 40 numbered cards in full pack. Since one card has to be drawn it can be done in ⁴⁰C₁ ways .

So the required probability will be ¹³C₁ x ⁴⁰C₁ / ⁵²C₂ = 13*40 / 1326 = 0.3921= 39.21% chance to get atleast one heart and one numbered card.