Probability Puzzle: Calculating Chances of Being Hit by Car After 10 Crossings

[rwinston]

Here is a probability question I came across recently:

If the chance of being hit by a car while crossing the street is 0.1, then what are the chances of being hit by a car after crossing the street 10 times?

Obviously the probability can't be 100% (the sum of the individual probabilities), so I think the answer must be something like the sum of the individual probabilities minus the probability that you get hit by a car on every combination of crossing the street - i.e. it is P(A) + P(B) + P(C) ... minus a combinatorial expansion of the probabilities of getting hit by a car on multiple street crossings.

Does anyone know the correct answer to this?

 

[Pere Callahan]

If we let aside the question of whether or not it makes sense to get hit more than one time, the probability of getting hit at least once is one minus the probability of not getting hit at all, that is
$$1-0.9^{10}\approx 0.65$$

 

[HallsofIvy]

First clarify you question. I might interpret "what are the chances of being hit by a car after crossing the street 10 times?" as "after I have crossed the street 10 times without getting hit by a car, what is the probability I will be hit crossing the 11th time."

Assuming the probability of getting hit one time is independent of the probability of getting hit a different time, the answer to that question is obviously 0.1!

If you mean "the probability of getting hit by a car at least once in 10 times", then Pere Callahan's answer is correct: subtract the chance of NOT getting hit in ten times, 0.9¹⁰, from 1.