# What is the probability of an occurrence within an interval

I am having an issue getting my head around this question as well. Gale-force winds occur on the average of 0.7 times every 54 hours. For an interval of 110 hours whats the probability r = 2?

I believe I should solve it with Poisson Distribution (as it is in this weeks lesson, and question references using two decimal places for lambda). I would like to know the methodology - not necessarily the answer, but if the answer was supplied it might help me work back and forth through the problem.

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Break the problem into steps. First, what's the rate per 110 hours? Second, given the rate, is that enough to compute Poisson probabilities? –  whuber Sep 28 '12 at 22:47
By the way, I think this is a terrible time to use a Poisson model. The fact that there is a hurricane season means the rate isn't constant. Depending on your definition, a single hurricane may generate several several gale force wind events clumped together more than you would expect from a Poisson model. You should know how to use the Poisson model, but I hope the next question is to reject it. –  Douglas Zare Sep 28 '12 at 23:45
I guess I am learning, are you suggesting the Poisson model is over applied, or just poorly applied in this overly academic example? –  akaphenom Sep 29 '12 at 2:34
I'm saying a Poisson distribution is probably a bad model for this particular example. –  Douglas Zare Sep 29 '12 at 3:58
I agree with @DouglasZare. This homework problem certainly has a Poisson model in mind, but it's not really the right model here due to the likely clumping. –  zbicyclist Sep 29 '12 at 5:03
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1. Calculate $\lambda$:
\eqalign{ 0.7/54 &= x/110 \\ &= 0.7 / 54 \times 110 \\ &\approx 1.43. }
2. Solve for $\Pr(r=2)$:
$$\Pr( r = 2 ) = e^{-\lambda} \frac{\lambda^r}{r!} = e^{-1.43}\frac{1.43^2}{2!} = 0.2443.$$
+1. A check, using R: dpois(2, 0.7 * 110 / 54) returns $0.2442824$. –  whuber Sep 29 '12 at 21:40