Week 1 - Lesson 4.7 Geometric Probability Distribution Flashcards
Like the Poisson and binomial distributions, this distribution describes a discrete random variable. Recall that in the binomial experiments, we tossed a coin a fixed number of times and counted the number, X,
of heads as successes.
Geometric probability distribution
This distribution describes a situation in which we toss the coin until the first head (success) appears. We
assume, as in the binomial experiments, that the tosses are independent of each other.
Geometric probability distribution
Characteristics of a Geometric Probability Distribution
• The experiment consists of a sequence of independent trials.
• Each trial results in one of two outcomes: success, S, or failure, F.
• The geometric random variable X is defined as the number of trials until the first S is observed.
• The probability p(x) is the same for each trial.
Why would we wait until a success is observed? One example is in the world of business. A business owner may
want to know the length of time a customer will wait for some type of service. Another example would be an
employer who is interviewing potential candidates for a vacant position and wants to know how many interviews
he/she has to conduct until the perfect candidate for the job is found. Finally, a police detective might want to know
the probability of getting a lead in a crime case after 10 people are questioned.
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The probability distribution, mean, and variance of a geometric random variable are given as follows:
formula on module
Example: A court is conducting a jury selection. Let X be the number of prospective jurors who will be examined
until one is admitted as a juror for a trial. Suppose that X is a geometric random variable, and p, the probability of a
juror being admitted, is 0.50.
a. Find the mean and the standard deviation of X.
b. Find the probability that more than two prospective jurors must be examined before one is admitted to the jury.
a. The mean and the standard deviation can be calculated as follows:
To find the probability that more than two prospective jurors will be examined before one is selected, you could
try to add the probabilities that the number of jurors to be examined before one is selected is 3, 4, 5, and so on, as
follows:
However, this is an infinitely large sum, so it is best to use the Complement Rule as shown:
In order to actually calculate the probability, we need to find p(1) and p(2). This can be done by substituting the
appropriate values into the formula:
Now we can go back to the Complement Rule and plug in the appropriate values for p(1) and p(2):
This means that there is a 0.25 chance that more than two prospective jurors will be examined before one is admitted
to the jury
Technology Note: Calculating Geometric Probabilities on the TI-83/84 Calculator
Press [2ND][DISTR] and scroll down to ’geometpdf(’. Press [ENTER] to place ’geometpdf(’ on your home screen.
Type in values of p and x separated by a comma, with p being the probability of success and x being the number of
trials before you see your first success. Press [ENTER]. The calculator will return the probability of having the first
success on trial number x.
Use ’geometcdf(’ for the probability of at most x trials before your first success.
Note: It is not necessary to close the parentheses.
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Characteristics of a Geometric Probability Distribution:
• The experiment consists of a sequence of independent trials.
• Each trial results in one of two outcomes: success, S, or failure, F.
• The geometric random variable X is defined as the number of trials until the first S is observed.
• The probability p(x) is the same for each trial.
The probability distribution, mean, and variance of a geometric random variable are given as follows:
answer on module
