I was planning my cycling schedule when I thought of this question... Can anyone explain why this is not true???
Suppose there is a 1% chance of a person getting knocked down by a vehicle each time he/she crosses the road. Then, the chances of a person being knocked down by a vehicle if he/she crosses 2 roads would be 2%. ... The chance of a person being knocked down will be 100% if he/she crosses 100 roads. But we know this is not true!
What's wrong with this logic????
That's not a paradox; it's bad math.
The chance of being knocked down when crossing two roads is: $1\% + 99\%\times 1\% = 1.99\%$
In general, let $N$ be the number of roads crossed until you are knocked down; then the probability of being knocked down at least once when trying to cross $n$ roads is: $$\mathsf P(N\leq n)=1-0.99^n$$
$$\mathsf P(N\leq 100)= 1-0.99^{100} \approx 63.4\%$$
However, the process is memoryless. The chance of being knocked down on crossing the one hundredth road, when given that you haven't been knocked down while crossing the first 99 roads is: $1\%$
$$\mathsf P(N\leq 100\mid N> 99) = \mathsf P(N=100) = 1\%$$