The point $P$ on a real number line is moved according to the rule as follows.
Flip a coin, if heads is the spot on the face, then $P$ is moved by $+1$, and if tails is the spot on the face, then $P$ is moved to the point of reflection for the origin. At first $P$ is set on the origin on the real number, let $a_n$ be the coordinate of $P$ after $n$ tossing the coin.

(1) What's the probability such that $a_3=0?$

(2) What's the probability such that $a_4=1?$

(3) Express the probability such that $a_n=n-3$ for $n\geq 3$ in terms of $n$.
Let's solve each part of the problem step-by-step.

### Part (1): Probability that \( a_3 = 0 \)

1. **Initial Position**: \( P \) starts at the origin, so \( a_0 = 0 \).
2. **Possible Moves**: Each coin flip can result in either \( +1 \) or a reflection through the origin.
3. **Sequence Analysis**: We need to find the sequences of moves that result in \( a_3 = 0 \).

   - **Heads (H)**: Move \( +1 \)
   - **Tails (T)**: Reflect through the origin

   To end up at \( 0 \) after 3 moves, we need to consider the parity and the effect of reflections:
   - One possible sequence is \( T, H, T \):
     - After the first T, \( P \) is still at \( 0 \).
     - After the second H, \( P \) moves to \( +1 \).
     - After the third T, \( P \) reflects back to \( 0 \).
   - Another possible sequence is \( H, T, H \):
     - After the first H, \( P \) moves to \( +1 \).
     - After the second T, \( P \) reflects to \( -1 \).
     - After the third H, \( P \) moves to \( 0 \).

   These are the only sequences that result in \( a_3 = 0 \).

4. **Probability Calculation**:
   - There are \( 2^3 = 8 \) possible sequences of 3 coin flips.
   - There are 2 sequences that result in \( a_3 = 0 \).

   Therefore, the probability is:
   \[
   \frac{2}{8} = \frac{1}{4}
   \]

### Part (2): Probability that \( a_4 = 1 \)

1. **Initial Position**: \( P \) starts at the origin, so \( a_0 = 0 \).
2. **Possible Moves**: Each coin flip can result in either \( +1 \) or a reflection through the origin.
3. **Sequence Analysis**: We need to find the sequences of moves that result in \( a_4 = 1 \).

   - To end up at \( 1 \) after 4 moves, we need to consider the parity and the effect of reflections:
   - One possible sequence is \( H, H, T, H \):
     - After the first H, \( P \) moves to \( +1 \).
     - After the second H, \( P \) moves to \( +2 \).
     - After the third T, \( P \) reflects to \( -2 \).
     - After the fourth H, \( P \) moves to \( -1 \).

   - Another possible sequence is \( H, T, H, H \):
     - After the first H, \( P \) moves to \( +1 \).
     - After the second T, \( P \) reflects to \( -1 \).
     - After the third H, \( P \) moves to \( 0 \).
     - After the fourth H, \( P \) moves to \( +1 \).

   These are the only sequences that result in \( a_4 = 1 \).

4. **Probability Calculation**:
   - There are \( 2^4 = 16 \) possible sequences of 4 coin flips.
   - There are 2 sequences that result in \( a_4 = 1 \).

   Therefore, the probability is:
   \[
   \frac{2}{16} = \frac{1}{8}
   \]

### Part (3): Probability that \( a_n = n - 3 \) for \( n \geq 3 \)

1. **Initial Position**: \( P \) starts at the origin, so \( a_0 = 0 \).
2. **Possible Moves**: Each coin flip can result in either \( +1 \) or a reflection through the origin.
3. **Sequence Analysis**: We need to find the sequences of moves that result in \( a_n = n - 3 \).

   - To end up at \( n - 3 \) after \( n \) moves, we need to consider the parity and the effect of reflections:
   - The number of reflections must be odd (by parity) and it is at most 3, so the only possibilities are 1 and 3.
   - If there is one reflection, it has to be the second move.
   - If there are three reflections, the first has to be the first move, and the other two have to be consecutive.

4. **Probability Calculation**:
   - The total number of sequences is \( 2^n \).
   - The number of valid sequences is \( n - 1 \).

   Therefore, the probability is:
   \[
   \frac{n - 1}{2^n}
   \]

The final answer is \( \boxed{ \frac{n - 1}{2^n} } \)