Probability problems(Deck of cards): find the probability (as a percentage) of d

Question

Probability problems(Deck of cards):

find the probability (as a percentage) of drawing any of the following poker hands in 5-card stud poker:

1) a three-of-a-kind

2) a four-of-a-kind

3) a "dead man's hand," which is a two-pairs hand consisting of the pair of black aces, the pair of black eight's along with an unknown fifth card (this is said to be the hand that Old Wild West gunfighter Will Bill Hickok's had when he was murdered playing poker..)

4) any two-pairs hand containing a pair of aces and a pair of eights

5) a flush, including straight flushes and royal flushes

6) a flush, excluding straight flushes and royal flushes

7) a straight, including flushes and royal flushes

8) a straight, excluding any flushes

Explanation / Answer

Solution:-

1) Percentage of  three of a kind = 2.1128%.

This hand has the pattern AAABC where A, B, and C are from distinct kinds.

The number of such hands = 13C1 × 4C3 ×12C2 × (4C1)2 = 54,912

Total number of combinations of different hands = 52C5 = 2,598,960

Probability of three of a kind = 54,912/2,598,960 = 0.021128.

Percentage of  three of a kind = 2.1128%.

2) Percentage of Four of a kind = 0.0240%.

This hand has the pattern AAAAB where A and B are from distinct kinds.

The number of such hands= 13C1 × 4C4 × 12C1 × 4C1 = 624

Total number of combinations of different hands = 52C5 = 2,598,960

Probability of four of a kind = 624/2,598,960 = 0.000240.

Percentage of Four of a kind = 0.0240%.

5) Percentage of flush is 0.198%.

Here all 5 cards are from the same suit

The number of such hands =4C1 × 13C5 = 5148

Total number of combinations of different hands = 52C5 = 2,598,960

Probability of flush = 5148/2,598,960 = 0.00198079.

Percentage of flush is 0.198%.

8) Percentage of straight is 0.3925%.

Here all 5 cards are from the same suit

The number of such hands = (10C1 × (4C1)5 ) - 40= 10,200

Total number of combinations of different hands = 52C5 = 2,598,960

Probability of straight = 10200/2598960 = 0.003925

Percentage of straight is 0.3925%

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