Calculating the Probability of Drawing a Number Card, Ace, and a Black Queen from a Deck

To calculate the probability of drawing a number card (2-10), an ace, and a black queen from a standard 52-card deck, we’ll walk through the calculations step-by-step. This method will help us understand the chances of such an event occurring and how to verify the result using Python code.

Calculating the Probability by Hand

1. Choosing the Number Card:

We need to choose one card from the 94 (9 number cards per suit times 4 suits) cards that are number cards (2-10). However, since a deck has only 36 distinct number cards (9 per suit times 4 suits), the number cards we can select are 36. Therefore, we have 36 choices.

2. Choosing the Ace:

We also need to choose one ace from the 4 aces available. This is simply 4 choices.

3. Choosing the Black Queen:

There are only 2 black queens (Queen of Spades and Queen of Clubs) in a deck. So, we have 2 choices.

4. Combination Formula:

The total number of ways to choose 3 distinct cards from 52 cards where the order doesn’t matter is given by the combination formula (binom{52}{3}), which equals (52 times 51 times 50 / (3 times 2 times 1) 22,100).

5. Total Combinations:

The total number of favorable outcomes is the product of the choices for each card type: (36 times 4 times 2 288).

6. Calculating Probability:

The probability (P) of drawing a number card, an ace, and a black queen is the ratio of the number of favorable outcomes to the total number of possible outcomes:

(P frac{288}{22,100} approx 0.0130) or 1.30%.

Verifying the Calculation with Python

To verify our hand calculation, let’s use Python. We’ll write a function to check if a card is a number card, an ace, or a black queen, and then use list comprehension to filter all possible combinations.

```python def isANumber(i): x i - 13 if x 1 or x 10: return True else: return False def isABlackQueen(i): x i - 13 if (i 26 or i 51) and x 11: return True else: return False def isAnAce(i): x i - 13 if x 0: return True else: return False l [ [i, j, k] for i in range(52) for j in range(52) for k in range(52) if i ! j and i ! k and j ! k if isANumber(i) or isANumber(j) or isANumber(k) if isABlackQueen(i) or isABlackQueen(j) or isABlackQueen(k) if isAnAce(i) or isAnAce(j) or isAnAce(k) ] print(len(l) / (52 * 51 * 50 / (3 * 2 * 1))) ``` The above Python code helps to verify the calculation. It will list all possible combinations and count the number of favorable outcomes. When running the code, the output should approximately match our hand calculated probability.

Conclusion

By following the steps above, we can easily calculate the probability of an event in a deck of cards. For drawing a number card (2-10), an ace, and a black queen, the probability is approximately 1.30%. This method can be extended to other similar problems in probability and combinatorics.

Related Keywords

- Probability - Deck of Cards - Combinations