Question

Miles draws three cards at random from a standard deck of 52 cards, without replacement.

(a) Find the probability that all three cards are red.

(b) Find the probability that all three cards have the same rank.

Answer 1

P( red, red, red no replacement ) = 2/17

P( 3 cards with same rank, no replacement) = 1/425

Explanation:

There are 52 cards, 26 are red

P( red) = red/total =26/ 52 = 1/2

Now draw the second card without replacement

There are 51 cards, 25 are red

P( red) = red/total =25/51

Now draw the third card without replacement

There are 50 cards, 24 are red

P( red) = red/total =24/50 = 12/25

P( red, red, red no replacement ) = 1/2 * 25/51 * 12/25 = 2/17

P ( all have the same rank)

There are 52 cards, 4 of each rank

We don't care what the first rank is, now the second and third draw have to have the same rank as the first

P( rank) = 1/1

Now draw the second card without replacement

There are 51 cards, 3 are left with the same rank

P( same rank) = cards with same rank/total = 3/51

Now draw the third card without replacement

There are 50 cards, 2 are left with the same rank

P( same rank) = cards with same rank/total = 2/50 = 1/25

P( 3 cards with same rank, no replacement) = 1 * 3/51*1/25 = 1/425