Question

The tree diagram shows the sample space of​ two-digit numbers that can be created using the digits 2​, 9​, 3​, and 7. What is the probability of choosing a number from the sample space that contains both 9 and 7​?
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Part 1
​P(9 and 7​)equals
  
enter your response here or
  
enter your response here​%
There are
  
enter your response here possible outcomes that include both 9 and 7. There are
  
enter your response here possible​ outcomes, which are equally likely.
​(Type integers,​ fractions, or mixed​ numbers.)   

Answer 1

To determine the probability of choosing a two-digit number that contains both 9 and 7 using the digits 2, 9, 3, and 7, we need to first identify the total possible outcomes and then the favorable outcomes that include both digits.

### Step 1: Identify Total Outcomes
When creating a two-digit number using the digits 2, 9, 3, and 7, each place (tens and units) can be filled by any of the 4 digits. Therefore, the total number of two-digit combinations is:

\[
4 \text{ (choices for the first digit)} \times 4 \text{ (choices for the second digit)} = 16
\]

### Step 2: Identify Favorable Outcomes
Next, we look for outcomes that include both 9 and 7. The two-digit combinations can be:
1. 97
2. 79

These are the only two two-digit numbers that can be formed with both 9 and 7.

### Step 3: Calculate the Probability
Now we calculate the probability \( P(9 \text{ and } 7) \):

\[
P(9 \text{ and } 7) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{16} = \frac{1}{8}
\]

If you want to express the probability as a percentage:

\[
P(9 \text{ and } 7) = \frac{1}{8} \times 100\% = 12.5\%
\]

### Final Answers
So the answers to the questions are:
- **Number of possible outcomes that include both 9 and 7:** 2
- **Total possible outcomes:** 16
- **Probability \( P(9 \text{ and } 7) \):** \( \frac{1}{8} \) or **12.5%**