Unizor - Probability - Advanced Problems 2

Problem A

A poker dealer gave you five cards from the standard deck of 52 cards.
What is the probability of getting "Four of a Kind" combination?
The combination "Four of a Kind" is the one when you have four cards of the same rank (say, four 9's or four King's) among five cards in your hand with the fifth card being of a different rank.

Answer: 13·48/ C[52,5]

Problem B

A poker dealer gave you five cards from the standard deck of 52 cards.
What is the probability of getting "Full House" combination?
The combination "Full House" is the one when you have three cards of one rank (say, three 8's or three Jack's) and two cards of another rank (say, two queens or two 5's) among five cards in your hand.

Answer: 13·C[4,3]·12·C[4,2] / C[52,5]

Problem C

A poker dealer gave you five cards from the standard deck of 52 cards.
What is the probability of getting "Straight" combination?
A "Straight" is a combination of five cards with sequential ranks of not of the same suit. For example, Queen, Jack, 10, 9, 8 of not of the same suit. If they are of the same suit, the combination is called "Straight Flush" and is considered to be a different combination.
There is one more rule for a "Straight". The Ace of any suit can be the highest ranking card in a deck for a "Straight" combination Ace, King, Queen, Jack, 10 or the lowest ranking card in a deck with the rank of 1 in a combination 5, 4, 3, 2, A
.
Answer: (10·4^5 − 10·4) / C[52,5]

Problem D

A poker dealer gave you five cards from the standard deck of 52 cards.
How many different "Three of a Kind" combinations you can get?
The combination "Three of a Kind" is the one when you have three cards of the same rank (say, three 9's or three Kings) among five cards in your hand, and the other two cards of not of this rank (otherwise, you would have "Four of a Kind" combination) and not of the same rank among themselves (otherwise, you would have "Full House" combination).

Answer: (13·C[4,3]·C[12,2]·4^2) / C[52,5]