2017-11-06_21-57-10_Geometric equations

Crystal Ball parameter restrictions

Geometric(*p*) models the total number of trials that will occur before the first success in a set of binomial trials, given that *p* is the probability of a trial succeeding. The total number of trials is equal to the total number of failures plus 1 (the first success). Examples of the Geometric distribution are shown below:

#### Examples

##### Cards

I select a card from a pack (no jokers) and guess its suit before looking at it. The total number of guesses I will have to get it right can be estimated as Geometric(25%). The Geometric distribution assumes that *p* is constant with each trial i.e. that I cannot get any better at guessing with each failure, nor does my problem change (so I'll have to put the card back and reshuffle). It also assumes that I will doggedly carry on, even if it takes me a hundred failures before I succeed. Thus, some caution is needed in its application.

##### Dry oil wells

The Geometric distribution is sometimes quoted as useful to estimate the total number of wells an oil company will drill in a particular section before getting a producing well. The total number of wells is equal to all dry wells plus the 1 producing well. That would, however, be assuming that a) the company does not learn from its mistakes; and b) it has the money and obstinacy to keep drilling new wells despite the cost.

##### More sensible example

You need to purchase some item, conduct a test or operation on that item, and if you fail, go and buy another. For example, you need to find a cow with disease X, but a definitive test involves an expensive procedure, so you randomly select a cow, test it, etc. The number of cows you'll need to buy is equal to Geometric(p) where p is the prevalence of disease X among the cows.

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Note that this would not work if you bought cows in batches. For example, if you bought batches of 5 cows. Then the number you'd have to buy to get an infected cow is: =(Geometric(P))*5, where P = 1-(1-p)5, i.e. the probability that a batch of 5 cows contains at least one infected.

#### Comments

The Geometric distribution is a special case of the 2017-11-06_21-56-47_ Negative Binomial for *s* = 1 i.e. Geometric(*p*) = NegBinomial(p, 1), which means that the sum of *s* independent Geometric(*p*) distributions = NegBinomial(p,*s*). The Geometric distribution is the discrete analogue of the 2017-11-06_21-56-52_ Exponential distribution, and gets its name because its probability mass function is a geometric progression. The Geometric distribution is occasionally called a *Furry distribution*.

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