[10 points] Problem 2: Two cards are randomly selected from a deck of 52 playing cards, What

	is the probability that both cards are greater than 3 and less than 6?

To solve the problem, we first need to determine how many cards in a standard 52-card deck are greater than 3 and less than 6. The cards that fit this criterion are the 4s and 5s.

In a standard deck of 52 cards, there are:
- 4 cards of 4 (one in each suit: hearts, diamonds, clubs, spades)
- 4 cards of 5 (one in each suit: hearts, diamonds, clubs, spades)

Thus, there are \(4 + 4 = 8\) cards that are greater than 3 and less than 6.

We are to find the probability that both cards drawn from the deck meet this criterion. Since the cards are drawn without replacement, the calculation involves a combination of probabilities for dependent events.

First, we calculate the probability that the first card drawn is one of these 8 cards:
\[ P(\text{First card is 4 or 5}) = \frac{8}{52} \]

Next, if the first card is indeed one of the desired cards, there are now 7 such cards remaining out of a total of 51 remaining cards. Hence, the probability that the second card drawn is also one of these cards is:
\[ P(\text{Second card is 4 or 5} \mid \text{First card is 4 or 5}) = \frac{7}{51} \]

To find the overall probability of both events occurring, we multiply the two probabilities together:
\[ P(\text{Both cards are 4 or 5}) = \frac{8}{52} \times \frac{7}{51} \]

Simplifying this expression:
\[ \frac{8}{52} = \frac{2}{13} \]
\[ \frac{7}{51} \text{ remains as is} \]

Thus, the combined probability is:
\[ P(\text{Both cards are 4 or 5}) = \frac{2}{13} \times \frac{7}{51} = \frac{14}{663} \]

Therefore, the probability that both cards selected are greater than 3 and less than 6 is:
\[ \boxed{\frac{14}{663}} \]

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. What is the probability of drawing two cards both of which are aces from a standard deck?
2. How many ways can you draw two cards from a deck such that one is a heart and the other is a spade?
3. What is the probability that the first card drawn is a face card (Jack, Queen, King) and the second card drawn is a number card?
4. If you draw three cards from a deck, what is the probability that all three are diamonds?
5. How does the probability change if the cards are drawn with replacement instead of without?

**Tip:** When dealing with probabilities involving drawing cards without replacement, always account for the reduction in the total number of cards as well as the specific number of desired cards remaining after each draw.

Two cards are randomly selected from a deck of 52 playing cards, What is the probability that both cards are greater than 3 and less than 6?

Learn how to calculate the probability of drawing two cards that are greater than 3 and less than 6 from a standard 52-card deck. This problem involves understanding dependent events and basic probability, making it suitable for students in Grades 9-12. The solution includes a step-by-step breakdown using the formula for dependent probabilities.

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