An ordinary deck of playing cards has 4 suits, with 13 cards of each suit. In many games, each of 4 players is dealt 13 cards at random.

Question

a. What is the probability that such a hand has each of the following?
1. Exactly 5 spades

2. Which is more probable, getting all 4 aces or getting all 13 cards of the same suit? Justify your answer.

Answer 1

let S be the event of being dealt a spade,
x the event of a non-spade

One such 5 spade hand could be
SSSSSXXXXXXXX
The prob of that is
(13/52*12/51*11/50*10/49*9/48)(39/47*38/46*37/45*36/44*35/43*34/33*33/32*32/31)

a quicker way would be to say
prob(exactly 5 spades)
= ( C(13,5)*C(39,8)/C(52,13)
= .12469

prob (4 aces)
= 4C(4,4)*C(48,9)/C(52,13) = appr .01056
prob(13 of same suit)
= 4/C(52,3) = very small
not even close!