Question
The tree diagram shows the sample space of two-digit numbers that can be created using the digits 5, 4, 3, and 7. What is the probability of choosing a number from the sample space that contains both 4 and 7?
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Part 1
P(4 and 7)equals
enter your response here or
enter your response here%
There are
enter your response here possible outcomes that include both 4 and 7. There are
enter your response here possible outcomes, which are equally likely.
(Type integers, fractions, or mixed numbers.)
LOADING... Click the icon to view the tree diagram.
Question content area bottom
Part 1
P(4 and 7)equals
enter your response here or
enter your response here%
There are
enter your response here possible outcomes that include both 4 and 7. There are
enter your response here possible outcomes, which are equally likely.
(Type integers, fractions, or mixed numbers.)
Answers
GPT-4o mini
To solve this problem involving the probability of choosing a two-digit number containing both 4 and 7 from a sample space created by the digits 5, 4, 3, and 7, we need to first determine the total number of two-digit outcomes possible and then identify how many of these contain both digits 4 and 7.
### Step 1: Determine Possible Outcomes
The two-digit numbers can be formed by using the digits 5, 4, 3, and 7, where each digit can be used in the tens place and the units place.
- For the **tens place**, you can choose any of the 4 digits (5, 4, 3, or 7).
- For the **units place**, you can also choose any of the 4 digits (5, 4, 3, or 7).
Thus, the total number of two-digit numbers is calculated as follows:
\[
\text{Total outcomes} = 4 \text{ (choices for tens)} \times 4 \text{ (choices for units)} = 16
\]
### Step 2: Identify Outcomes Containing Both 4 and 7
Next, we need to find the specific two-digit numbers that contain both 4 and 7. The combinations for our two-digit numbers that would meet this criteria are:
1. **Tens digit is 4** and **units digit is 7**: this gives us the number **47**.
2. **Tens digit is 7** and **units digit is 4**: this gives us the number **74**.
These are the only numbers that contain both 4 and 7.
### Counting Outcomes
Now we can count the outcomes that include both 4 and 7:
- There are **2 outcomes**: {47, 74}.
### Step 3: Calculate Probability
To find the probability \( P(4 \text{ and } 7) \):
\[
P(4 \text{ and } 7) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{16} = \frac{1}{8}
\]
### Step 4: Convert to Percentage
To convert this probability into percentage form:
\[
P(4 \text{ and } 7) \approx 0.125 = 12.5\%
\]
### Final Responses
So we have:
- Number of possible outcomes containing both 4 and 7: **2**
- Total possible outcomes: **16**
- Probability \( P(4 \text{ and } 7) \): **\(\frac{1}{8}\)** or **12.5%**
Here are the final answers you could use:
- \( P(4 \text{ and } 7) = \frac{1}{8} \) or \( 12.5\% \)
- There are **2** possible outcomes that include both 4 and 7.
- There are **16** possible outcomes, which are equally likely.
### Step 1: Determine Possible Outcomes
The two-digit numbers can be formed by using the digits 5, 4, 3, and 7, where each digit can be used in the tens place and the units place.
- For the **tens place**, you can choose any of the 4 digits (5, 4, 3, or 7).
- For the **units place**, you can also choose any of the 4 digits (5, 4, 3, or 7).
Thus, the total number of two-digit numbers is calculated as follows:
\[
\text{Total outcomes} = 4 \text{ (choices for tens)} \times 4 \text{ (choices for units)} = 16
\]
### Step 2: Identify Outcomes Containing Both 4 and 7
Next, we need to find the specific two-digit numbers that contain both 4 and 7. The combinations for our two-digit numbers that would meet this criteria are:
1. **Tens digit is 4** and **units digit is 7**: this gives us the number **47**.
2. **Tens digit is 7** and **units digit is 4**: this gives us the number **74**.
These are the only numbers that contain both 4 and 7.
### Counting Outcomes
Now we can count the outcomes that include both 4 and 7:
- There are **2 outcomes**: {47, 74}.
### Step 3: Calculate Probability
To find the probability \( P(4 \text{ and } 7) \):
\[
P(4 \text{ and } 7) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{16} = \frac{1}{8}
\]
### Step 4: Convert to Percentage
To convert this probability into percentage form:
\[
P(4 \text{ and } 7) \approx 0.125 = 12.5\%
\]
### Final Responses
So we have:
- Number of possible outcomes containing both 4 and 7: **2**
- Total possible outcomes: **16**
- Probability \( P(4 \text{ and } 7) \): **\(\frac{1}{8}\)** or **12.5%**
Here are the final answers you could use:
- \( P(4 \text{ and } 7) = \frac{1}{8} \) or \( 12.5\% \)
- There are **2** possible outcomes that include both 4 and 7.
- There are **16** possible outcomes, which are equally likely.