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:
- 97
- 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%