Respuesta :
Question:
A 5-card hand is dealt from a perfectly shuffled deck of playing cards.
What is the probability that the hand is a two of a kind?
A two of a kind has two cards of the same rank (called the pair). Among the remaining three cards, not in the pair, no two have the same rank and none of them have the same rank as the pair. For example, {4♠, 4♦, J♠, K♣, 8♥} is a two of a kind.
Answer:
P(two of a kind) = 42.3%
Step-by-step explanation:
The probability that the hand is a two of a kind is given by
P(two of a kind) = No. of ways to produce two of a kind/Total no. of ways to deal 5-hand cards
There are total 52 cards in a standard deck of playing cards.
Total number of ways to deal 5-card hand is given by
Total number of ways = ₅₂C₅
Total number of ways = 2595960
So there are 2595960 different ways of dealing 5-card hands
Now we will find out the number of ways to produce two of a kind.
The number of ways to select the rank of two matching cards is given by
Rank of matching cards = ₁₃C₁ = 13
Since the matching cards must be of same rank.
The number of ways to select the rank of remaining 3 cards is given by
Rank of remaining 3 cards = ₁₂C₃ = 220
Since the remaining ranks are now 12.
The number of ways to select the suits of two matching cards is given by
Suits of two matching cards = ₄C₂ = 6
The number of ways to select the suits of 1st non-matching card is given by
Suits of 1st non-matching card = ₄C₁ = 4
The number of ways to select the suits of 2nd non-matching card is given by
Suits of 2nd non-matching card = ₄C₁ = 4
The number of ways to select the suits of 3rd non-matching card is given by
Suits of 3rd non-matching card = ₄C₁ = 4
Finally, the probability is
P(two of a kind) = No. of ways to produce two of a kind/Total no. of ways to deal 5-hand cards
P(two of a kind) = (₁₃C₁ × ₁₂C₃ × ₄C₂ × ₄C₁ × ₄C₁ × ₄C₁) / ₅₂C₅
P(two of a kind) = (13 × 220 × 6 × 4 × 4 × 4) / 2595960
P(two of a kind) = 1098240/2595960
P(two of a kind) = 0.423
P(two of a kind) = 42.3%
