Geometric Probability Distributions
Defining Geometric Probability Distributions
A geometric 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.
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
Xis defined as the number of trials until the first Sis 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.
Probability Distribution, Mean, and Variance of a Geometric Random Variable
The probability distribution, mean, and variance of a geometric random variable are given as follows:
Finding the Mean, Standard Deviation, and Probability
The mean and the standard deviation can be calculated as follows:
b. Find the probability that more than two prospective jurors must be examined before one is admitted to the jury.
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:
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.
Calculating the Probability of Success
Using the geometric distribution with a success probability of 0.4, calculate the probability of getting your first success on the third trial.
The probability that the first success is on the third trial is 0.144.
Technology Notes: Calculating Geometric Probabilities on the TI-83/84 Calculator
Note: It is not necessary to close the parentheses.
A venture capitalist invests in start-up companies in Silicon Valley, California. Each start-up company either succeeds or fails. If a company fails the venture capitalist loses $3 million dollars; if the company succeeds the capitalist gains $8 million dollars. What is the probability that the investor will fail with the first 11 companies and succeed for the first time on his/her 12th investment? What assumption do you have to make to determine this probability?
Matthew is a high school basketball player and a 75% free throw shooter. What is the probability that Matthew makes his first free throw on his fifth shot?
The probability that the first success is on the third trial is approximately 0.003.
- A prison reports that the number of escape attempts per month has a Poisson distribution with a mean value of 1.5.
- Calculate the probability that exactly three escapes will be attempted during the next month.
- Calculate the probability that exactly one escape will be attempted during the next month.
- The mean number of patients entering an emergency room at a hospital is 2.5. If the number of available beds today is 4 beds for new patients, what is the probability that the hospital will not have enough beds to accommodate its new patients?
- An oil company has determined that the probability of finding oil at a particular drilling operation is 0.20. What is the probability that it would drill four dry wells before finding oil at the fifth one? (Hint: This is an example of a geometric random variable.)
- Suppose the probability of a high school senior working full-time when in college is .234 and suppose you randomly select one senior until you find one who expects to work full-time while in college. You are interested in the number of seniors you must ask.
- In words, define the random variable.
- What is the distribution of this random variable?
- Construct the probability function for
X. Stop at X=6.
- On average, how many seniors would you expect to have to ask until you found one who expects to work full-time when in college?
- What is the probability that you will have to ask fewer than 4 high school seniors?
- Construct a histogram for this distribution.
- What are the four conditions for the geometric probability setting?
- Explain the difference between the binomial probability distribution and the geometric probability distribution.
Xhas a geometric distribution with probability of success p, what does (1−p)(n−1)represent?
- What does the expected value of a geometric random variable represent?
- You play a game that you can either win or lose. Your probability of winning is .38. What is the probability that it takes 6 games until you win?
- Suppose you are looking for a friend to go to the movies with you. The probability that a friend will agree to go with you is .27. What is the probability that the fifth friend you ask will be the first one to agree to go with you?
- Given a geometric probability distribution with probability of success .3. Compute each of the following:
P(X=6) P(X>4) P(X≤7) P(X>9) P(X≥8) P(3≤X≤10) P(3<X<10)
To view the Review answers, open this PDF file and look for section 4.7.