Questions: Three cards are drawn with replacement from a standard deck of 52 cards. Find the probability that the first card will be a club, the second card will be a red card, and the third card will be a queen. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Solution Steps

To solve this problem, we need to calculate the probability of drawing a specific sequence of cards from a standard deck of 52 cards with replacement. The probability of each event is independent due to replacement. We calculate the probability of each event separately and then multiply them together to find the overall probability.

1. Probability of drawing a club: There are 13 clubs in a deck of 52 cards, so the probability is \( \frac{13}{52} \).
2. Probability of drawing a red card: There are 26 red cards (hearts and diamonds) in a deck, so the probability is \( \frac{26}{52} \).
3. Probability of drawing a queen: There are 4 queens in a deck, so the probability is \( \frac{4}{52} \).

Finally, multiply these probabilities together to get the overall probability.

The probability of drawing a club from a standard deck of 52 cards is given by the ratio of the number of clubs to the total number of cards. There are 13 clubs in the deck.

\[ P(\text{Club}) = \frac{13}{52} = 0.25 \]

The probability of drawing a red card (either a heart or a diamond) is given by the ratio of the number of red cards to the total number of cards. There are 26 red cards in the deck.

\[ P(\text{Red}) = \frac{26}{52} = 0.5 \]

The probability of drawing a queen is given by the ratio of the number of queens to the total number of cards. There are 4 queens in the deck.

\[ P(\text{Queen}) = \frac{4}{52} \approx 0.07692 \]

Since the cards are drawn with replacement, the events are independent. Therefore, the overall probability of drawing a club first, a red card second, and a queen third is the product of the individual probabilities.

\[ P(\text{Club, Red, Queen}) = P(\text{Club}) \times P(\text{Red}) \times P(\text{Queen}) = 0.25 \times 0.5 \times 0.07692 \approx 0.009615 \]

The probability can also be expressed as a fraction in its lowest terms.

\[ P(\text{Club, Red, Queen}) = \frac{1}{104} \]

Final Answer