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.

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.

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%

the clues that signal the reader should change tone are the punctuation marks, the grammatical signs. for example, the quotation marks at the beginning of something someone else said literally or the exclamation marks.

the tone of the first line of dialogue until "gettysburg" is a kind tone, a tone of advise. the narrator is trying to the other person in doing something he or she obviously is finding hard to do by giving a piece of advise and bringing up a memory of a successful similar case.

the clue that the reader understand how to read the word "bang" is the exclamation mark. it gives the word a surprise tone, a strong accent.

the best tone for reading the word "bang" is an exciting tone, a surprise one, even a loud one.

the words that should be read with a formal tone are the ones that give factual information. the sentence: mister lincoln couldn't think of anything to say at the gettysburg" gives information about an event and it needs to be read formally, also, when the narrator wants to transmit calmness, a formal and slow tone is needed, because people also transmit messages with the vibrations of our voices and tones.

step-by-step explanation: