PSDRV - Probability Models: Binomial and Geometric Probability Distributions Lesson
Probability Models: Binomial and Geometric Probability Distributions Lesson
There are two very special probability distributions. Frequently we encounter situations where there are only TWO outcomes of interest like tossing a coin to yield heads or tails, attempting a free throw in basketball which will either be successful or not, predicting the sex of an unborn child (either male or female), quality testing of manufactured items which will either meet requirements or not. In each case we can describe the two outcomes as either a success or a failure depending on how the experiment is defined.
When four specific conditions are satisfied in an experiment it is called a BINOMIAL setting which will produce a BINOMIAL PROBABILITY DISTRIBUTION.
The four requirements are:
a) each observation falls into one of two categories called a success or failure
b) the observations are all independent of each other
c) the probability of success (p) for each observation is the same - equally likely
d) there is a fixed number of observations or trials
These requirements have a special name: BERNOULLI TRIALS.
Statistics jargon: If the experiment is a binomial setting, then the random variable X represents the number of successes and is called a binomial random variable, and the probability distribution of X is called a binomial distribution. This variable will assume a finite number of positive integer values and represents a special member of the general family of discrete random variables. The next unit will discuss random variables in more detail.
BINOMIAL DISTRIBUTION DEFINED
The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n and is written as X = B(n, p).
The second special probability distribution is a Geometric distribution. In this scenario there are a fixed number of successes (ONE! - the FIRST) and counts the number of trials needed to obtain that first success. It is theoretically possible to proceed indefinitely without ever obtaining a success. Because of this requirement these problems are often referred to as "waiting" problems since you are waiting for something to occur.
Examples would be:
1) flip a coin UNTIL you get your first head
2) roll a die UNTIL you get your first 3
3) attempt a three-point shot in basketball UNTIL you make your first basket
A random variable X is geometric provided that the following conditions are met: (conditions a-c are the same as for the binomial probability distribution)
a) each observation falls into one of just two categories, called success or failure
b) the observations are all independent of each other
c) the probability of success (p) for each observation is the same - equally likely
NEW
d) the variable of interest is the number of trials required to obtain the FIRST success.
Recognizing the existence of either a binomial or geometric distribution is essential to knowing how to proceed with your data analysis. Here is an example that should help explain how to VERIFY a geometric setting.
Example
An experiment consists of rolling a single die. The event of interest is rolling a 3 which is called a success. The random variable is defined as X = number of trials UNTIL a 3 occurs. To VERIFY that this is a geometric setting, note that rolling a 3 will represent a success and rolling any other number will represent a failure. The probability of rolling a 3 on each roll is the same at 1/6. The observations are independent because each roll of the die does not affect the next roll. A trial consists of rolling the die once. We roll the die until a 3 appears. Since all of the requirements are satisfied, this experiment describes a geometric setting.
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