Questions: You draw one card from a 52-card deck. Then the card is replaced in the deck and the deck is shuffled, and you draw again. Find the probability of drawing a three the first time and a heart the second time. The probability of drawing a three the first time and a heart the second time is (Type an integer or a simplified fraction.)

Solution Steps

To find the probability of drawing a three the first time and a heart the second time, we need to calculate the probability of each independent event and then multiply them together. The probability of drawing a three from a 52-card deck is the number of threes (4) divided by the total number of cards (52). Similarly, the probability of drawing a heart is the number of hearts (13) divided by the total number of cards (52). Since the card is replaced and the deck is shuffled, these events are independent.

The probability of drawing a three from a standard 52-card deck is calculated by dividing the number of threes (4) by the total number of cards (52). Thus, the probability is:

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

The probability of drawing a heart from a 52-card deck is calculated by dividing the number of hearts (13) by the total number of cards (52). Thus, the probability is:

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

Since the events are independent (the card is replaced and the deck is shuffled), the combined probability of drawing a three first and a heart second is the product of the individual probabilities:

\[ P(\text{Three and Heart}) = P(\text{Three}) \times P(\text{Heart}) = 0.07692 \times 0.25 = 0.01923 \]

Final Answer