Math Problem Statement

Let a point P move on a straight line according to the score shown on a fair dice that we throw by the following rules: If the score is 6, P returns to the origin O. If the score is 1, 2, or 3, P moves 1 in a positive direction. If the score is 4 or 5, P moves 1 in a negative direction. What is the probability that after four throws of the dice, P is at the origin?

Solution

The problem described in the image involves a random walk on a straight line, determined by the outcome of rolling a fair six-sided dice. Here's a summary of the given rules:

  • Score = 6: The point PP returns to the origin OO.
  • Score = 1, 2, 3: The point PP moves +1+1 in a positive direction.
  • Score = 4, 5: The point PP moves 1-1 in a negative direction.

The task is to determine the probability that after four throws of the dice, the point PP is at the origin OO.

Approach:

To solve this, we need to analyze the possible moves:

  • Moving in the positive direction occurs with a probability of 36=12\frac{3}{6} = \frac{1}{2} (scores 1, 2, or 3).
  • Moving in the negative direction occurs with a probability of 26=13\frac{2}{6} = \frac{1}{3} (scores 4 or 5).
  • Returning to the origin occurs with a probability of 16\frac{1}{6} (score 6).

We are interested in the probability that after four dice rolls, the point PP ends up back at the origin. For this to happen, the net movement must be zero, meaning the number of moves in the positive direction must cancel out the number of moves in the negative direction, while taking into account that any occurrence of a "6" brings the point directly to the origin.

To solve this problem, I will calculate the required probability. Let me proceed with the steps.The probability that the point PP is at the origin after four dice throws is 23\frac{2}{3} or approximately 0.6667.

Let me know if you would like a detailed explanation of the calculations or if you have any further questions.

Here are some related questions to explore this topic further:

  1. What would be the probability if the number of throws was increased to 6?
  2. How would the outcome change if we used a different dice with varying probabilities for each move?
  3. Can you generalize this solution for any number of throws?
  4. What is the expected final position of point PP after four rolls of the dice?
  5. How does the inclusion of the "return to origin" condition affect the overall probability?

Tip: Random walk problems often involve balancing positive and negative movements, which is key to calculating probabilities of returning to a starting point.

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Math Problem Analysis

Mathematical Concepts

Probability
Random Walk
Combinatorics

Formulas

P(origin after 4 throws) = Sum of favorable outcomes / Total possible outcomes

Theorems

Law of Total Probability

Suitable Grade Level

Grades 9-12