Notes
Practice Problems
You are playing tennis with a friend. The probability you win each game is 40%.
\(\textbf{1)}\) What is the probability your first win is on the 3rd game?
\(\textbf{2)}\) What is the probability your first win is on the 5th game?
\(\textbf{3)}\) What is the probability your first win is on the 1st game?
\(\textbf{4)}\) What is the mean number of games until your first win?
You are rolling a fair 6-sided die. The probability you roll a 5 is \(\frac{1}{6}\).
\(\textbf{5)}\) What is the probability your first “5” is on the 3rd roll?
\(\textbf{6)}\) What is the probability your first “5” is on the 5th roll?
\(\textbf{7)}\) What is the probability your first “5” is on the 1st roll?
\(\textbf{8)}\) What is the mean number of rolls until your first “5”?
See Related Pages\(\)
\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Uniform Distribution}\)
\(\,\,\,\,\,\,\,\,p(x)=\frac{1}{b-a}…\)
\(\bullet\text{ Binomial Distribution}\)
\(\,\,\,\,\,\,\,\,p(r)={}_{n}C_{r}(p)^r(1-p)^{n-r}…\)
\(\bullet\text{ Poisson Distribution}\)
\(\,\,\,\,\,\,\,\,P(x)=\displaystyle\frac{\lambda^x e^{-\lambda}}{x!}…\)
\(\bullet\text{ Continuity Correction}\)
\(\,\,\,\,\,\,\,\,c-.5\lt x\lt c+.5…\)
In Summary
The geometric distribution is a statistical model that describes the number of failures that occur before the first success in a series of independent and identical Bernoulli trials. It is used in fields such as engineering, computer science, and economics.
Real world examples of Geometric Distribution
The number of times a customer must call a company before their issue is resolved: Each time a customer calls, there is a fixed probability that their issue will be resolved on the first call. If it is not resolved, they must call again and try their luck again, creating a geometric distribution.
The number of times a machine must be used before it fails: Each time a machine is used, there is a fixed probability that it will fail on that use. If it does not fail, it must be used again and the process is repeated, creating a geometric distribution.
The number of times a gambler must play a game of chance before winning: Each time a gambler plays a game of chance, there is a fixed probability that they will win on that play. If they do not win, they must play again and the process is repeated, creating a geometric distribution.
The number of times a salesperson must make a cold call before making a sale: Each time a salesperson makes a cold call, there is a fixed probability that they will make a sale on that call. If they do not make a sale, they must call again and the process is repeated, creating a geometric distribution.
The number of times a scientist must conduct an experiment before obtaining a successful result: Each time a scientist conducts an experiment, there is a fixed probability that they will obtain a successful result on that trial. If they do not obtain a successful result, they must conduct the experiment again and the process is repeated, creating a geometric distribution.
Topics that use geometric distribution
Reliability engineering: In reliability engineering, the geometric distribution is often used to model the number of failures or defects in a system before the first success or acceptable unit is observed. For example, a manufacturer might use the geometric distribution to model the number of defective products that are produced before the first non-defective product is produced on a production line.
Queueing theory: In queueing theory, the geometric distribution is often used to model the number of customers that arrive at a service facility before the first customer is served. For example, a bank might use the geometric distribution to model the number of customers that arrive at a teller window before the first customer is served.
Epidemiology: In epidemiology, the geometric distribution is often used to model the number of secondary infections caused by a single infected individual in a population. For example, a public health agency might use the geometric distribution to model the number of individuals that are infected by a single infected individual in an outbreak of a contagious disease.
Marketing: In marketing, the geometric distribution is often used to model the number of contacts with a customer before a sale is made. For example, a company might use the geometric distribution to model the number of marketing emails that are sent to a customer before the customer makes a purchase.