used to find the probability
of success after a certain number of trials
, in which "success" is defined as occurance of a certain event
. If X
is a random variable
representing the number of independent
trials until success, and p
is the fixed probability of success for each trial, then we say X
). The probability density function
is given as follows:
P(X = x) = (x) = (1 - p)x-1p for x = 1, 2, ...
P(X = x) = (x) = 0 otherwise
With the geometric distribution, the mean and population variance are given by μ = 1/p and σ2 = (1 - p)/p2 respectively.
The origin of the pdf is relatively easy to understand. The probability of success on the xTH trial is equal to (x - 1) failures (each with probability of (1 - p)) and one success (with probability p). So multiply the two probabilities and you get (1 - p)x-1p.
Example: Suppose the probability that a randomly selected E2 writeup is deleted by dannye is 0.015. Find the probability that none of your next 100 writeups have a date with dannye's "kill" button.
Let X be a random variable for the number of writeups you create before dannye kills one. Assume X ~ Geometric(0.015)
P(X ≥ 100) = P(X = 100) + P(X = 101) + P(X = 102) + ...
P(X ≥ 100) = 0.985990.015 + 0.9851000.015 + 0.9851010.015 + ...
P(X ≥ 100) = 0.98598(0.015)(0.985 + 0.9852 + 0.9853 + ...)
* Recall that z + z2 + z3 + ... is the geometric series, and it approximates z/(1-z). Thus:
P(X ≥ 100) = 0.98598(0.015)(0.985/0.015) = 0.224
So, there's a 22.4% chance that dannye won't kill one of your next 100 writeups!