MHB What Is the Probability That a Bridge Is Empty at Noon?

[anemone]

Here is this week's POTW:

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It takes 5 minutes to cross a certain bridge and 1000 people cross it in a day of 12 hours, all times of day being equally likely. Find the probability that there will be nobody on the bridge at noon.

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[anemone]

Congratulations to castor28 for his correct solution (and special thanks to his other approach to solve the same problem worded slightly different than the original one(Cool)), which you can find below:

Solution from castor28:

Spoiler

The bridge will be empty at noon if no visitors arrive between 11:55 and noon (assuming that interval is included in the opening hours).

There are $144$ intervals of $5$ minutes in the 12-hours period. The probability that none of the $1000$ visitors arrive in that interval is therefore $\left(\dfrac{143}{144}\right)^{1000}\approx 0.00094090411913581$.

We may also consider the slightly different problems of a continuous process where visitors arrive at an average rate of $1000/144\approx 6.944\ldots$ visitors per 5 minutes. In that case, the probability is given by the Poisson distribution and is equal to $e^{-\frac{1000}{144}}\approx 0.000963975725734177$. This is a slightly different value; this comes from the fact that the number of visitors on a single day will not necessarily be exactly equal to $1000$, unlike in the original problem.