Probability of Visiting Grandparents

Problem

Probability of Visiting Grandparents, problem

Solution 1

The schedules of the trains in the two directions are shifted relative to each other:

Probability of Visiting Grandparents, solution

If, e.g., on the red line, the trains arrive at $3, 6, 9,$ etc. minutes to an hour, while on the blue line that's $1, 4,7,10$ minutes, etc. then the chances of the boy catching the red direction are $2$ to $1.$

Solution 2

The pdf (probability distribution function) of his arrival is such that the ratio of the areas under the pdf for time intervals corresponding to the two direction is ~$2.$

Solution 3

Probability of Visiting Grandparents, solution 3

Acknowledgment

This is an old problem that I heard of yet in high school. Can't pinpoint the source. Solution 2 is by Amit Itagi; Solution 3 is by Elia Noris.

 

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