If 5 cards are randomly selected from a standard deck of 52 cards without, repla
ID: 3203460 • Letter: I
Question
If 5 cards are randomly selected from a standard deck of 52 cards without, replacement, find the probabilities of Getting two pair (exactly two cards of one rank and two cards of another rank, such as J J 4 4 9v). The two pairs cannot match each other and the fifth card cannot be of either rank. Getting four of a kind (a hand containing four cards of the same rank and one card of another rank, such as 9 9 9 9 J). Getting a full house (a hand containing three cards of one rank and two cards of another rank, such as 3 3 3 6 6)Explanation / Answer
Solution:-
a) Probability of two pairs = 0.047539.
This hand has the pattern AABBC where A, B, and C are from distinct kinds.
The number of such hands = 13C2 × 4C2 × 4C2 ×11C1 × 4C1 = 123,552
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of two pairs = 123,552/2,598,960 = 0.047539
b)Probability of Four of a kind = 0.000240.
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.
c) Probability of getting a full house = 0.001441.
This hand has the pattern AAABB where A and B are from distinct kinds.
The number of such hands = 13C1 × 4C3 × 12C1 × 4C2. = 3774
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of getting a full house = 3774/2,598,960 = 0.001441.