Slide1 : Geometric Distributions have to follow these 4 conditions:
Each observation has only 2 possibilities: success or failure.
Each observation is independent of the others.
The probability of success, p, is the same for each observation.
The variable of interest is the number of trials required to obtain the first success. Chapter 8.2 - Geometric Distributions
The difference between binomial and geometric distributions is the number of trials needed. : The difference between binomial and geometric distributions is the number of trials needed.
Another real life example is the game of craps:
Once you roll your point, for example, an 8, you keep rolling until you get your 8 (win) or you roll a 7 (lose). In binomial distributions, there are a fixed number of trials (ie you have 3 children) In geometric distributions, you keep trying until the event you want to occur happens (ie you keep having children until a boy is born)
Slide3 : Now, how does this apply to probability?
Well, if you let p represent the probability of success, and 1-p the probability of failure, then: P(X = k) = (1-p)n-1 p This makes perfect sense – you keep trying n-1 times, and each time is a failure (probability value is1-p), and then the last time you succeed (probability value is p)
Slide4 : Let’s try an example:
If your probability of shooting a free throw and making it is .7 (70%), then the probability of missing is .3 What is the probability that you make a basket on your 3rd shot? P(X=3) = (.3)2 (.7) = .09 * .7 = .063 On the calculator:
Geometpdf(p,X) which is under 2nd VARS (DISTR) menu
So for this example, geometpdf(.7,3) = .063 What this means: miss on the 1st shot and miss on the 2nd shot and make the 3rd shot.
And means multiply.
So this becomes .3 * .3 * .7