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Chapter 7: Problem 40

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? Two pairs

Short Answer

Expert verified
The probability of being dealt a two-pair hand in a 5-card poker hand is approximately \(0.0475\) or \(4.75\%\) when calculated as: \(P(\text{two-pair}) = \frac{\binom{13}{2} \cdot \binom{4}{2} \cdot \binom{4}{2} \cdot \binom{44}{1}}{\binom{52}{5}}\).

Step by step solution

01

Calculate the number of ways to select the pairs.

We have 13 ranks of cards (from 2 to Ace). We need to choose 2 ranks for our pairs. This can be done in \(\binom{13}{2}\) ways.
02

Calculate the number of ways to select the cards for each pair.

For each selected rank from Step 1, there are 4 cards of that rank in the deck (one of each suit). We need to choose 2 cards out of 4 for each of the 2 ranks, which can be done in \(\binom{4}{2}\) ways for each rank. Since there are 2 ranks forming pairs, the total number of ways to select the individual cards for both pairs is \(\binom{4}{2} \cdot \binom{4}{2}\).
03

Calculate the number of ways to select the fifth card.

After selecting the two pairs, there are 11 remaining ranks and 4 suits for each rank. Thus, there are 44 cards left to choose from for the fifth card. We need to choose 1 card out of these 44, which can be done in \(\binom{44}{1}\) ways.
04

Calculate the total number of ways to form a two-pair hand.

To find the total number of ways to form a two-pair hand, we must multiply the results from Steps 1, 2, and 3. So, the total number of two-pair hands is \(\binom{13}{2} \cdot \binom{4}{2} \cdot \binom{4}{2} \cdot \binom{44}{1}\).
05

Calculate the total number of possible 5-card poker hands.

There are 52 cards in a deck, and we need to choose 5 of them to create a poker hand. This can be done in \(\binom{52}{5}\) ways.
06

Calculate the probability of getting a two-pair hand.

Now that we have the total number of ways to form a two-pair hand and the total number of possible 5-card poker hands, we can calculate the probability by dividing the two-pair hands by the total poker hands: \(P(\text{two-pair}) = \frac{\binom{13}{2} \cdot \binom{4}{2} \cdot \binom{4}{2} \cdot \binom{44}{1}}{\binom{52}{5}}\).
07

Simplify and find the probability.

Compute the values in the expression above, and then simplify: \(P(\text{two-pair}) = \frac{\binom{13}{2} \cdot \binom{4}{2} \cdot \binom{4}{2} \cdot \binom{44}{1}}{\binom{52}{5}} = \frac{78 \cdot 6 \cdot 6 \cdot 44}{2,598,960} = \frac{123,552}{2,598,960} \approx 0.0475\). The probability of being dealt a two-pair hand in a 5-card poker hand is approximately 0.0475 or 4.75%.

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Most popular questions from this chapter

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