To find the probability that all five cards drawn from a pack of 52 playing cards are number cards and there are at least two cards of the same number, we need to follow these steps:

Step 1: Total Possible Outcomes

The total number of ways to draw 5 cards from 52 is given by the combination: $\binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960$

Step 2: Total Number Cards

There are 40 number cards in a deck (A, 2, 3, ..., 10 in each of the 4 suits).

Step 3: Ways to Draw All Number Cards

The number of ways to draw 5 cards from these 40 number cards is: $\binom{40}{5} = \frac{40!}{5!(40-5)!} = 658,008$

Step 4: Ways to Draw 5 Number Cards with No Repetition

The number of ways to draw 5 cards from the 40 number cards with no repetition (all different ranks) is calculated by selecting 5 different ranks from the 10 available, and then selecting one of the 4 suits for each: $\binom{10}{5} \times 4^5 = 252 \times 1024 = 258,048$

Step 5: Ways to Draw 5 Number Cards with At Least One Repetition

The number of ways to draw 5 number cards with at least one repetition is: $\text{Total ways to draw all number cards} - \text{Ways to draw all different ranks}$ $658,008 - 258,048 = 399,960$

Step 6: Probability Calculation

The probability is the ratio of the number of favorable outcomes (drawing 5 number cards with at least one repetition) to the total number of ways to draw 5 cards from 52: $P(\text{5 number cards, at least one repetition}) = \frac{399,960}{2,598,960}$ $= \frac{399,960}{2,598,960} = \frac{9999}{64974} \approx 0.1538$

Conclusion

The probability that all five cards drawn are number cards and there are at least two cards of the same number is approximately $0.1538$ or $15.38\%$.

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Related Questions

1. What is the probability of drawing exactly two cards of the same number from the 40 number cards?
2. How many ways can you draw five cards from a deck such that all cards are face cards?
3. What is the probability that all five cards drawn are face cards?
4. How many ways can you draw 5 cards such that you get at least one Ace?
5. What is the probability that all five cards drawn are from the same suit?
6. How many combinations are there to draw five cards that include at least one King?
7. What is the probability of drawing four cards of the same rank and one card of a different rank?
8. How many ways are there to draw five cards that include at least one Jack and one Queen?

Tip

When dealing with probabilities involving multiple conditions, break the problem into smaller parts and calculate the probabilities step by step.