To solve the problem step by step, we will first determine the total number of two-digit numbers that can be formed using the digits 2, 3, 5, 7, and 9 without repetition. Then, we will find the probabilities for the two scenarios: forming an odd number and forming a multiple of 5.
### Step 1: Calculate the total number of two-digit numbers
1. **Choose the first digit (tens place)**: We can choose any of the 5 digits (2, 3, 5, 7, 9) for the tens place. This gives us 5 options.
2. **Choose the second digit (units place)**: After selecting the first digit, we have 4 remaining digits to choose from for the units place.
Thus, the total number of two-digit numbers formed is:
\[
\text{Total numbers} = 5 \times 4 = 20
\]
### Step 2: Calculate the probability of forming an odd number
1. **Identify the odd digits**: The odd digits available are 3, 5, and 9.
2. **Choose the units place**: We can place any of the 3 odd digits in the units place.
3. **Choose the tens place**: After placing an odd digit in the units place, we have 4 remaining digits to choose from for the tens place.
Thus, the total number of odd two-digit numbers formed is:
\[
\text{Odd numbers} = 3 \times 4 = 12
\]
4. **Calculate the probability**: The probability of forming an odd number is given by the ratio of the number of favorable outcomes (odd numbers) to the total outcomes (total two-digit numbers):
\[
P(\text{odd number}) = \frac{\text{Number of odd numbers}}{\text{Total numbers}} = \frac{12}{20} = \frac{3}{5}
\]
### Step 3: Calculate the probability of forming a multiple of 5
1. **Identify the digit for multiples of 5**: The only digit that can be in the units place to form a multiple of 5 is 5.
2. **Choose the units place**: We can place 5 in the units place (1 way).
3. **Choose the tens place**: After placing 5 in the units place, we can choose any of the remaining 4 digits (2, 3, 7, 9) for the tens place.
Thus, the total number of two-digit numbers that are multiples of 5 is:
\[
\text{Multiples of 5} = 1 \times 4 = 4
\]
4. **Calculate the probability**: The probability of forming a multiple of 5 is given by:
\[
P(\text{multiple of 5}) = \frac{\text{Number of multiples of 5}}{\text{Total numbers}} = \frac{4}{20} = \frac{1}{5}
\]
### Final Answers
1. The probability that the number formed is an odd number is \( \frac{3}{5} \).
2. The probability that the number formed is a multiple of 5 is \( \frac{1}{5} \).
---
