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Suppose there is an event that happens with a probability of y in x interval of time, what would be the probability of it happening in x/2 interval of time? Would that be y/2 or is there something more to be considered?

To be a bit more specific, why do use exponential probability in this case:

The probability of a car passing a certain intersection in a 20 minute windows is 0.9. What is the probability of a car passing the intersection in a 5 minute window? (Assuming a constant probability throughout)

The answer is computed using the logic that the probability of a car not passing in 20 mts = (probability of a car not passing in 5 mts)^4. While that makes logical sense why is it exponentiation and not something as directly as (probability of car passing in 20 mts/4)? I find the latter more intuitive.

I believe there is something fundamental that I am missing here. Am I assuming that the probability curve is |_|_|_| where the bar indicates the exact time a car passes by as opposed to ----- where the probability is just uniform i.e no time instance has any significant edge over the other?

Nishant
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    It would depend on the situation. For example, your probability of surviving in a shark tank over 1 hour is 5%. Your survival chance would not drop to 2.5% if you had to stay in the tank for 30 minutes, just half of the original time. – RK01 Jul 28 '15 at 13:20
  • It would have to be a perfect situation in which x and y are linearly correlated. There aren't too many cases of this happening. – RK01 Jul 28 '15 at 13:26
  • But the probability of my spotting a shark would become 2.5% right? Ok suppose sharks are uniform in the water - that is the linear correlation you are talking about? – Nishant Jul 28 '15 at 13:28
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    Mathematically speaking, yes, but in the real world, the rate is exponential because sharks will take some time to detect you, so they may be dormant in the first 30 minutes. That's why I said that the situation has to be perfect. – RK01 Jul 28 '15 at 13:30
  • To add a real world example, suppose there is a probability that a car passes a point 50% in 10 mts, assuming cars are evenly distributed, I have 25% of seeing a car in 5 mts - is there any flaw in that reasoning? That was my original real world problem so just trying to relate. And in the first comment, the mathematical probability is 10% right? I suspect its something just inverse. – Nishant Jul 28 '15 at 13:34
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    The only flaw I see occurs when cars pass at intervals less than 5 minutes. Then, the probability for both would be at 100%. – RK01 Jul 28 '15 at 13:36
  • I think this is different from discreet math which I am familiar with, which branch of probability should I refer to learn these concepts better? – Nishant Jul 28 '15 at 13:41

1 Answers1

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It depends on the distribution. If you say that an event happens with probability $y$ in an interval $x$, I guess what you mean is that the probability that the event happens within the first $x$ time units is $y$. In other words, the probability that the time instance $t$ at which the event happens is within $[0,x]$.

It thus depends on the distribution of the time instance $t$ at which the event happens. If we assume that the time instance is always positive (i.e., that $t$ is a positive random variable), then we can say that the probability that the event happens before $x$ is equal to $F_t(x)$, where $F_t$ is the cumulative distribution function of the random variable $t$.

Now, in your question you say that $y=F_t(x)$ and you ask in which case you have $y/2=F_t(x/2)$. For a fixed $x$, this might happen for many random variables. But if you want this property to hold for all $x$ within a given range, then the uniform distribution is the only correct solution.

Bernhard
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