1. **Problem statement:** We have a point moving on a number line starting at 0. When a die is rolled: - If the result is 1 or 2, the point moves 1 step to the left (negative direction). - If the result is 3, 4, 5, or 6, the point moves 1 step to the right (positive direction). 2. **Given:** - After 1 roll, the probability the point is at position $-1$ is $p$. - After 2 rolls, the probability the point is at position $0$ is $p$. - After 3 rolls, the point can be at 3 different positions. 3. **Goal:** Find the position $r$ where the probability $\Lambda(r)$ is the greatest. 4. **Step 1: Define probabilities for one roll.** - Probability of moving left (positions $-1$): $\frac{2}{6} = \frac{1}{3}$. - Probability of moving right (positions $+1$): $\frac{4}{6} = \frac{2}{3}$. 5. **Step 2: After 1 roll:** - Possible positions: $-1$ and $+1$. - $P(-1) = \frac{1}{3}$, $P(1) = \frac{2}{3}$. 6. **Step 3: After 2 rolls:** - Possible positions: $-2, 0, 2$. - Calculate $P(0)$ (the point returns to 0): $$P(0) = P(-1) \times P(\text{right}) + P(1) \times P(\text{left}) = \frac{1}{3} \times \frac{2}{3} + \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}.$$ 7. **Step 4: After 3 rolls:** - Possible positions: $-3, -1, 1, 3$. - Calculate probabilities: - $P(-3) = P(-2) \times P(\text{left}) = ?$ (from previous step, $P(-2) = P(-1) \times P(\text{left}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$) - So, $P(-3) = \frac{1}{9} \times \frac{1}{3} = \frac{1}{27}$. - Similarly, calculate $P(1)$ and $P(3)$. 8. **Step 5: General observation:** - The point is more likely to move right because $P(\text{right}) = \frac{2}{3}$ is greater than $P(\text{left}) = \frac{1}{3}$. - Therefore, the position with the highest probability after multiple rolls will be shifted to the right. 9. **Step 6: Conclusion:** - The position $r$ with the highest probability $\Lambda(r)$ is the one shifted right the most, i.e., the maximum positive position reachable after the given number of rolls. **Final answer:** The position $r$ where $\Lambda(r)$ is maximum is the positive integer corresponding to the number of right moves minus left moves, which is the most probable position shifted to the right.