Probability of drawing two pairs from a standard deck of cards

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Eli
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#1

Question:

Five cards are randomly drawn from a standard deck of cards. Find the probability of obtaining two pairs.
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#2

Is this solution correct ?

To find the probability of obtaining two pairs, we can break down the problem into a few steps.

Step 1: Choose the ranks of the two pairs.

There are 13 different ranks in a standard deck of cards, so we have 13 choices for the first pair. After picking the first pair, there are 12 remaining ranks to choose from for the second pair. Therefore, the number of ways to choose the ranks of the two pairs is given by

\[\binom{13}{1} \times \binom{12}{1}\].

Step 2: Choose the cards for the first pair.

Each pair consists of two cards, so we need to choose 2 cards from 4 cards of the chosen rank for the first pair. This can be done in \(\binom{4}{2}\) ways.

Step 3: Choose the cards for the second pair.

Similar to step 2, we also need to choose 2 cards from 4 cards of the chosen rank for the second pair. This can also be done in \(\binom{4}{2}\) ways.

Step 4: Choose the remaining card.

Once the two pairs are chosen, we have one remaining card to select. This can be any of the remaining 44 cards in the deck.

Step 5: Calculate the total number of ways to obtain two pairs.

To find the total number of ways to obtain two pairs, we multiply the results from each step:

\[\text{Total number of ways} = \binom{13}{1} \times \binom{12}{1} \times \binom{4}{2} \times \binom{4}{2} \times 44 \]

Step 6: Calculate the total number of possible selections.

When selecting 5 cards from a standard deck of 52 cards, the total number of possible selections is given by \(\binom{52}{5}\) .

Step 7: Calculate the probability of obtaining two pairs.

Finally, we can calculate the probability by dividing the total number of ways to obtain two pairs by the total number of possible selections:

\[\text{Probability} = \frac{\text{Total number of ways}}{\text{Total number of possible selections}}\]

NB

In a standard deck of cards, the ranks refer to the different values or numbers assigned to each card. In a standard 52-card deck, the ranks include Ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), and King (K). Each suit (hearts, diamonds, clubs, and spades) consists of cards with the same ranks. So, when we talk about choosing the ranks of the two pairs, we are referring to selecting two different numbers or values from the available ranks in the deck.
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