1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?

Question

A hypothetical example: > You have a 1/1000 chance of being hit by a bus when crossing the street. However, if you perform the action of crossing the street 1000 times, then your chance of being hit by a bus increases to about 60% because every time you do the action, the probability of it happening again increases. <!-- additional text for minimum post --> What is the math behind this to support this? Just curious.

Answer 1

If there is a 1/1000 chance each time, then there is a 999/1000 chance each time that the action does not occur.

If the experiment is repeated n times, and the odds to do not change, then odds that the outcome never occurs is exactly (0.999)ⁿ.

Thus the probability that the accident does occur at least once must be 1 - (0.999)ⁿ

^{That turns out to be about 0.62.3 or 63.2 % by direct calculation using n = 1000. The value is also approximately 1 - 1/e}