To solve the problem step by step, we need to calculate the probability of drawing an even number first, an odd number second, and a number divisible by 3 third. We will follow these steps:
### Step 1: Calculate the probability of drawing an even number on the first draw.
- **Total numbers from 1 to 50**: 50
- **Even numbers from 1 to 50**: 2, 4, 6, ..., 50 (which are 25 even numbers)
The probability of drawing an even number is given by:
\[
P(\text{Even}) = \frac{\text{Number of Even Numbers}}{\text{Total Numbers}} = \frac{25}{50} = \frac{1}{2}
\]
### Step 2: Calculate the probability of drawing an odd number on the second draw.
- **Odd numbers from 1 to 50**: 1, 3, 5, ..., 49 (which are also 25 odd numbers)
The probability of drawing an odd number is given by:
\[
P(\text{Odd}) = \frac{\text{Number of Odd Numbers}}{\text{Total Numbers}} = \frac{25}{50} = \frac{1}{2}
\]
### Step 3: Calculate the probability of drawing a number divisible by 3 on the third draw.
- **Numbers divisible by 3 from 1 to 50**: 3, 6, 9, ..., 48.
To find how many numbers are divisible by 3, we can use the formula for the nth term of an arithmetic sequence:
- First term (a) = 3
- Common difference (d) = 3
- Last term (l) = 48
To find the number of terms (n):
\[
l = a + (n-1)d \implies 48 = 3 + (n-1) \cdot 3
\]
\[
48 - 3 = (n-1) \cdot 3 \implies 45 = (n-1) \cdot 3 \implies n-1 = 15 \implies n = 16
\]
Thus, there are 16 numbers divisible by 3.
The probability of drawing a number divisible by 3 is given by:
\[
P(\text{Divisible by 3}) = \frac{\text{Number of Numbers Divisible by 3}}{\text{Total Numbers}} = \frac{16}{50} = \frac{8}{25}
\]
### Step 4: Calculate the combined probability of all three events.
Since the draws are independent (the slip is replaced each time), we multiply the probabilities:
\[
P(\text{Even, Odd, Divisible by 3}) = P(\text{Even}) \cdot P(\text{Odd}) \cdot P(\text{Divisible by 3})
\]
\[
= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{8}{25} = \frac{1 \cdot 1 \cdot 8}{2 \cdot 2 \cdot 25} = \frac{8}{100} = \frac{2}{25}
\]
### Final Answer:
The probability that an even number appears on the first draw, an odd number on the second draw, and a number divisible by 3 on the third draw is \(\frac{2}{25}\).
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