Question:
Five cards are randomly drawn from a standard deck of cards. Find the probability of obtaining two pairs.
-
- Active Topics
-
-
- by Eli 17 hours ago Re: What is in Your Mind? View the latest post Replies 703 Views 299748
- by Eli 17 hours ago Russia Invades Ukraine View the latest post Replies 663 Views 234898
- by Eli 2 days ago All in One: YouTube, TED, X, Facebook and Instagram Reels, Videos, Images and Text Posts View the latest post Replies 326 Views 33931
- by Eli 3 days ago PySpark for Large Data Processing View the latest post Replies 1 Views 6569
- by Eli 1 week ago President Museveni's Speech During International Development Association (IDA) Summit View the latest post Replies 1 Views 280
- by Eli 1 week ago From Simple Linear Regression Analysis to Covariance & Correlation to Independent Determinant, and R-Squared View the latest post Replies 11 Views 24832
- by Eli 1 week ago Collection of Greatest Christian Hymns of all Times View the latest post Replies 34 Views 67099
- by Eli 1 week ago Pondering Big Cosmology Questions Through Lectures and Dialogues View the latest post Replies 34 Views 58065
- by Eli 1 week ago Programmatically Manipulate Files: Renaming, Reading, Writing, Deleting, and Moving Files Between Folders View the latest post Replies 7 Views 15537
- by Eli 2 weeks ago Iran Launches Retaliatory Attack Against Israel, and Israel Retaliates by Attacking Iranian Isfahan Millitary Base View the latest post Replies 28 Views 17471
-
Probability of drawing two pairs from a standard deck of cards
- Eli
- Senior Expert Member
- Reactions: 183
- Posts: 5382
- Joined: 9 years ago
- Location: Tanzania
- Has thanked: 75 times
- Been thanked: 88 times
- Contact:
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.
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.
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
-
- Information
-
Who is online
Users browsing this forum: No registered users and 1 guest