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Thread: Probability of an event occurring 5 times??

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    Probability of an event occurring 5 times??




    Hi all

    I have recently placed a bet with a bookmaker concerning the number of days Sydney will exceed 35C in summer. Summer has 90 days and the long term average number of days exceeding 35C is 2.1 so the probability that any given day will reach 35C is 0.0233 or just over 2%.

    My problem is the bet is less than 6 days above 35C. So how do I calculate, assuming each day is independent (I know they are not truly independent given the weather passes through calendar days but for simplicity I will assume), the probability that there will be 5 or less days above 35C.

    Cheers, Tom

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    Re: Probability of an event occurring 5 times??

    I am away from my computer but you want to start by look at the binomial distribution with probability of event = 0.0233.

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    Re: Probability of an event occurring 5 times??

    As hlsmith points out the problem is binomial in nature and can be solved by using the binomial cumulative distribution function which has the formula:

    F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i}

    You can read up on the distribution and how to interpret the formula here. In plain language, the probability of at most five days reaching 35 degrees over the 90 days of summer at a 2.33% chance is the sum of the probability of it happening exactly 5 days + 4 days + 3 days + 2 days + 1 day + 0 days. In R we can quickly calculate those probabilities (from 5 to 0 days):

    Code: 
    > dbinom(5:0, 90, 0.0233)
    [1] 0.04068480 0.09915372 0.19109751 0.27308565 0.25724334 0.11981381
    The total of which is:

    Code: 
    pbinom(5, 90, .0233)
    [1] 0.9810788
    So, pretending that the observations are independent, there's a 98.11% probability that there will be at most five days above 35 days in a Sydney summer and, inversely, a 1.89% chance that there will 6 or more days above 35. But as you rightly point out, these estimates are not reliable as weather observations are not independent and, with a changing climate, future weather patterns are less likely to match past patterns. So, out of interest, what odds were you offered by the bookmaker?

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    Re: Probability of an event occurring 5 times??


    Thanks so much for your help. I have odds of 1.87. I don't think the bookie knows anything about the weather. 4 days down in summer and no days yet above 35C.

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