A standard deck of playing cards is shuffled and three people each choose a card. Find the probability that the first two cards chosen are hearts and the third card is red if the cards are chosen with replacement, and if the cards are chosen without replacement.

To solve this problem, we will consider two scenarios: one where the cards are chosen **with replacement** and another where the cards are chosen **without replacement**. We'll calculate the probability for each case.

### 1. With Replacement

When cards are chosen with replacement, each draw is independent of the previous draws. Thus, the probability remains the same for each draw.

#### Probability Calculations:

- **Probability the first card is a heart**:  
  A standard deck has 52 cards, with 13 hearts. Therefore, the probability of drawing a heart on the first draw is:
  \[
  P(\text{First card is a heart}) = \frac{13}{52} = \frac{1}{4}
  \]

- **Probability the second card is a heart**:  
  Since the card is replaced, the probability remains the same:
  \[
  P(\text{Second card is a heart}) = \frac{13}{52} = \frac{1}{4}
  \]

- **Probability the third card is red**:  
  There are 26 red cards (13 hearts and 13 diamonds) out of 52 cards. Thus:
  \[
  P(\text{Third card is red}) = \frac{26}{52} = \frac{1}{2}
  \]

#### Combined Probability:

The combined probability for the first two cards being hearts and the third card being red is the product of these probabilities:

\[
P(\text{First two hearts, third red}) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{2} = \frac{1}{32}
\]

### 2. Without Replacement

When cards are chosen without replacement, each draw affects the probabilities of subsequent draws.

#### Probability Calculations:

- **Probability the first card is a heart**:
  \[
  P(\text{First card is a heart}) = \frac{13}{52} = \frac{1}{4}
  \]

- **Probability the second card is a heart** (after the first heart is chosen, there are 12 hearts left out of 51 cards):
  \[
  P(\text{Second card is a heart | first was a heart}) = \frac{12}{51}
  \]

- **Probability the third card is red** (after two hearts are chosen, there are 24 red cards left out of 50 cards):
  \[
  P(\text{Third card is red | first two were hearts}) = \frac{24}{50} = \frac{12}{25}
  \]

#### Combined Probability:

The combined probability for the first two cards being hearts and the third card being red is:

\[
P(\text{First two hearts, third red}) = \frac{13}{52} \times \frac{12}{51} \times \frac{12}{25}
\]

Let's calculate this probability:

\[
P(\text{First two hearts, third red}) = \frac{13}{52} \times \frac{12}{51} \times \frac{12}{25} = \frac{1}{4} \times \frac{4}{17} \times \frac{12}{25} = \frac{48}{1700} = \frac{24}{850}
\]

\[
P(\text{First two hearts, third red}) \approx 0.0282
\]

### Summary of Results:

- **With replacement**: \(\frac{1}{32} = 0.03125\)
- **Without replacement**: \(\frac{24}{850} \approx 0.0282\)

Would you like further clarification or additional details on any of these steps?

### 5 Related Questions:

1. What is the probability that all three cards chosen are hearts without replacement?
2. How would the probabilities change if four people choose cards instead of three?
3. What is the probability that the first card chosen is a heart, the second a diamond, and the third a spade with replacement?
4. If the cards are chosen without replacement, what is the probability that the first two cards are black and the third is a face card?
5. What is the probability of drawing a king on each of three draws with replacement?

**Tip:** When solving probability problems involving replacement, remember that each event is independent, while without replacement, each event is dependent on the previous one.

Find the probability of drawing two hearts and one red card from a standard deck of 52 playing cards, both with and without replacement. This problem involves key probability concepts such as independent and dependent events, the multiplication rule, and conditional probability. Ideal for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation