A man takes a step forward with probability 0.7 and backward with probability 0.3. If the man takes 12 steps, and let the probability that he is just one step away from his initial position be p, then 13.23 - 15p is equal to ____

Answer

To solve the problem, we need to determine the probability \( p \) that the man is just one step away from his initial position after taking 12 steps.

### Step-by-step Solution:

1. **Understanding the Problem**:
The man can take a step forward with probability 0.7 and backward with probability 0.3. After 12 steps, we want to find the probability that he is either 1 step ahead or 1 step behind his starting position.

2. **Setting Up the Equations**:
Let \( x \) be the number of steps forward and \( y \) be the number of steps backward. The total number of steps taken is:
\[ x + y = 12 \]
The net position after these steps is:
\[ x - y = 1 \quad \text{(for being 1 step ahead)} \]
or
\[ x - y = -1 \quad \text{(for being 1 step behind)} \]

3. **Solving the Equations**:
- For \( x - y = 1 \):
\[ x + y = 12 \quad \text{(1)} \]
\[ x - y = 1 \quad \text{(2)} \]
Adding (1) and (2):
\[ 2x = 13 \implies x = 6.5 \quad \text{(not possible)} \]

- For \( x - y = -1 \):
\[ x + y = 12 \quad \text{(1)} \]
\[ x - y = -1 \quad \text{(2)} \]
Adding (1) and (2):
\[ 2x = 11 \implies x = 5.5 \quad \text{(not possible)} \]

4. **Conclusion**:
In both cases, we find that \( x \) is not an integer, which means that it is impossible for the man to be exactly 1 step away from his initial position after taking 12 steps. Therefore, the probability \( p \) is:
\[ p = 0 \]

5. **Calculating the Expression**:
Now we need to compute \( 13.23 - 15p \):
\[ 13.23 - 15 \times 0 = 13.23 \]

### Final Answer:
The value of \( 13.23 - 15p \) is \( 13.23 \).

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