To solve this, let’s break it down step-by-step.

1. Identify Total Possible Hands:

   • A 13-card hand is dealt from a standard 52-card deck.
   • The number of ways to choose any 13 cards from 52 is: $\binom{52}{13}$

2. Specific Favorable Hands:

   • We need exactly 3 red cards and 10 black cards in the hand.

   • There are 26 red cards and 26 black cards in the deck.

   • The number of ways to choose 3 red cards from 26 red cards is: $\binom{26}{3}$

   • The number of ways to choose 10 black cards from 26 black cards is: $\binom{26}{10}$

   • Therefore, the total number of favorable hands is: $\binom{26}{3} \times \binom{26}{10}$

3. Calculate the Probability:

   • The probability of getting exactly 3 red cards and 10 black cards is the ratio of the favorable hands to the total possible hands: $\frac{\binom{26}{3} \times \binom{26}{10}}{\binom{52}{13}}$

Now, let’s compute this:

• $\binom{26}{3} = 2600$
• $\binom{26}{10} = 5311735$
• $\binom{52}{13} = 635,013,559,600$

So, $\text{Probability} = \frac{2600 \times 5311735}{635013559600} = \frac{13810411000}{635013559600} \approx 0.0217$

Answer

The probability is approximately 0.0217, so the correct answer is: 0.0217

Would you like more details or have any questions?

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1. What is the probability of getting 5 red and 8 black cards?
2. How does the probability change if we draw 10 cards instead of 13?
3. What’s the probability of getting only red cards in a 13-card hand?
4. How many ways are there to draw 3 red and 10 black cards?
5. How would the solution change for a deck with jokers?

Tip: Using combinations simplifies probability calculations for card draws without replacement.