There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?
1. **Understanding the Problem:**
   - You are running on a race course for 1 hour.
   - You pass 300 runners going in the opposite direction.
   - Some of these runners are counted twice because you pass them twice.
   - We need to find the expected number of different runners you pass, not counting duplicates.

2. **Splitting the Problem into Two Cases:**
   - Since each lap takes 1 hour and you are on the course for exactly 1 hour, we can split the problem into two 30-minute intervals.
   - In the first 30 minutes, if you see a runner, there is a $\frac{1}{2}$ chance that you will see them again in the next 30 minutes.

3. **Calculating the Expected Number of Runners in the First 30 Minutes:**
   - You pass 300 runners in total.
   - Since the runners are uniformly distributed, you would expect to pass half of them in the first 30 minutes.
   - Therefore, the expected number of runners you pass in the first 30 minutes is $\frac{300}{2} = 150$.

4. **Calculating the Expected Number of Runners Seen Again:**
   - Out of these 150 runners, there is a $\frac{1}{2}$ chance of seeing each one again in the next 30 minutes.
   - Therefore, the expected number of runners you see again is $\frac{150}{2} = 75$.

5. **Calculating the Expected Number of New Runners in the Second 30 Minutes:**
   - In the second 30 minutes, you again pass 150 runners.
   - Out of these 150 runners, 75 are those you saw in the first 30 minutes.
   - Therefore, the expected number of new runners you see in the second 30 minutes is $150 - 75 = 75$.

6. **Summing Up the Expected Number of Different Runners:**
   - The total expected number of different runners you pass is the sum of the runners you see in the first 30 minutes and the new runners you see in the second 30 minutes.
   - Therefore, the expected number of different runners is $150 + 75 = 225$.

\[
\boxed{225}
\]