Crossing Bridge in Crowds

Problem

Crossing Bridge in Crowds, problem

Solution

If there is nobody on the bridge at noon, no one has entered it in the five-minute interval before noon. Since there are $144$ intervals of $5$ minutes in $12$ hours, the probability that an individual enters the bridge in a specific one is $\displaystyle\frac{1}{144}.$ The probability that none of the $1000$ individuals enters in that interval is

$\displaystyle\left(1-\frac{1}{144}\right)^{1000}=\left[\left(1-\frac{1}{144}\right)^{144}\right]^{\frac{1000}{144}}\approx e^{-\frac{125}{18}}\approx 0.00096.$

Acknowledgment

This is problem 68 from the Canadian Crux Mathematicorum (v1, 1975). The problem is by E.G. Dworschak, the solution is by G.D. Kaye.

 

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