The geometcdf( Command

Command Summary

Calculates the cumulative geometric probability for a single value

Command Syntax

geometcdf(probability, trials)

Press:

1. 2ND DISTR to access the distribution menu
2. ALPHA E to select geometcdf(, or use arrows.

Press ALPHA F instead of ALPHA E on a TI-84+/SE with OS 2.30 or higher.

TI-83/84/+/SE

2 bytes

This command is used to calculate cumulative geometric probability. In plainer language, it solves a specific type of often-encountered probability problem, that occurs under the following conditions:

1. A specific event has only two outcomes, which we will call "success" and "failure"
2. The event is going to keep happening until a success occurs
3. Success or failure is determined randomly with the same probability of success each time the event occurs
4. We're interested in the probability that it takes at most a specific amount of trials to get a success.

For example, consider a basketball player that always makes a shot with 1/4 probability. He will keep throwing the ball until he makes a shot. What is the probability that it takes him no more than 4 shots?

1. The event here is throwing the ball. A "success", obviously, is making the shot, and a "failure" is missing.
2. The event is going to happen until he makes the shot: a success.
3. The probability of a success - making a shot - is 1/4
4. We're interested in the probability that it takes at most 4 trials to get a success

The syntax here is geometcdf(probability, trials). In this case:

:geometcdf(1/4,4


This will give about .684 when you run it, so there's a .684 probability that he'll make a shot within 4 throws.

Note the relationship between geometpdf( and geometcdf(. Since geometpdf( is the probability it will take exactly N trials, we can write that geometcdf(1/4,4) = geometpdf(1/4,1) + geometpdf(1/4,2) + geometpdf(1/4,3) + geometpdf(1/4,4).

# Formulas

Going off of the relationship between geometpdf( and geometcdf(, we can write a formula for geometcdf( in terms of geometpdf(:

(1)
\begin{align} \operatorname{geometcdf}(p,n) = \sum_{i=1}^{n} \operatorname{geometpdf}(p,i) = \sum_{i=1}^{n} p\,(1-p)^{i-1} \end{align}

(If you're unfamiliar with sigma notation, $\sum_{i=1}^{n}$ just means "add up the following for all values of i from 1 to n")

However, we can take a shortcut to arrive at a much simpler expression for geometcdf(. Consider the opposite probability to the one we're interested in, the probability that it will not take "at most N trials", that is, the probability that it will take more than N trials. This means that the first N trials are failures. So geometcdf(P,N) = (1 - "probabiity that the first N trials are failures"), or:

(2)
\begin{align} \operatorname{geometcdf}(p,n) = 1-(1-p)^n \end{align}

.