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If there is an $80\%$ chance of rain in the next hour, what is the percentage chance of rain in the next half hour?


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Is that all the information you're given? Seems like there's not enough information here. – anegligibleperson Jul 7 '12 at 17:40
For another question on rain, see… – Ross Millikan Jul 7 '12 at 17:42

Hint: You have to specify your assumptions. The easiest one is that rain at any moment is independent of rain at any other moment and the same from moment to moment. Clearly, this is unrealistic, but it is the best we have. Then if the probability it doesn't rain in a given half hour is $p$, you need to successive independent not-rain events to have no rain in an hour. Does this help?

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Let $x$ be the probability that it will rain in next half an hour.
So $$\text{P(rain in 1st half) + P(rain in 2nd half)=P(rain in an hour)}$$ $$x+(1-x)x=\text{ probability of raining in 1 hr }=.8$$ $$x=1-\sqrt{.2}$$

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You could assume that occurrence of rain is a point process with constant rate lambda. Then the process is Poisson and the probability of rain in an interval of time [0,t] is given by the exponential distribution and if T = time until the event rain then

P[T<=t] = 1-exp(-λt) assuming λ is the rate per hour then P(rain in 1 hour)=1-exp(-λ).

and P(rain in 1/2 hour)=1-exp(-λ/2). Other models for the point process would give different answers.

In general the longer the time interval greater the chance of rain.

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