4.5: Geometric Distributions
Suppose you want to predict how many hands of blackjack it would take to win one hand. You could get lucky and win on the first hand or you might have to play two, three or even ten hands before you finally win. This is different from a binomial distribution because a binomial distribution has a limited and finite number of trials while a geometric distribution has a potentially unlimited number of trials. A binomial distribution also focuses on the number of successes that occur in n trials while a geometric distribution focuses instead on the number of trials before the first success.
Watch This
First watch this video to learn about geometric distributions.
CK12 Foundation: Chapter4GeometricDistributionsA
Then watch this video to see some examples.
CK12 Foundation: Chapter4GeometricDistributionsB
Guidance
The geometric distribution comes directly from our knowledge of the binomial distribution. The geometric distribution focuses on the number of trials before the first success occurs.
Suppose we play a game where we flip a coin until someone flips a tails and then the game is over. The probability that the game is over on the first flip is 50% because that is the probability that someone flips tails. After the first trial, if the game is not over then again the coin has 50% chance of getting a tails on the second flip. This means that the game has a 50% chance of being over on the first coin flip and a 25% chance of being over on the second coin flip.
The probability of performing \begin{align*}a\end{align*}
\begin{align*}P(a \ \text{trials})=q^{(a1)}\times p\end{align*}

\begin{align*}a\end{align*}
a is the number of trials ending in 1 success 
\begin{align*}p\end{align*}
p is the probability of success 
\begin{align*}q\end{align*}
q is the probability of failure
Example A
Consider the coin game where the game ends once someone flips tails. What is the probability that the game ends on the 3rd flip?
\begin{align*}a&=3 \ \text{because there are 3 trials}\\
q&=.5\\
p&=.5\end{align*}
\begin{align*}P(a \ \text{trials})&=q^{(a1)}\times p\\
P(3 \ \text{trials})&=\left ( \frac{1}{2}\right )^{(31)}\times \frac{1}{2}\\
P(3 \ \text{trials})&=\left (\frac{1}{2}\right )^2 \times \frac{1}{2}\\
P(3 \ \text{trials})&=.125\end{align*}
Example B
Suppose a copy machine consistently has a 5% chance of breaking on any given day. Although it might work for many days in a row, it will inevitably break down. What is the probability that it lasts a whole five day work week and breaks down on the 6th day?
Although it is counterintuitive “success” is defined in this problem to be the machine breaks down.
\begin{align*}p&=.05\\
q&=.95\end{align*}
\begin{align*}P(6 \ \text{trials})=.95^5\times .05=.039\end{align*}
Example C
Consider the coin game again and make a probability distribution for games lasting up to 8 coin flips.
a (number of flips)  Probability 

1 
\begin{align*}.5^0\times .5=.5\end{align*} 
2 
\begin{align*}.5^1\times .5=.25\end{align*} 
3 
\begin{align*}.5^2\times .5=.125\end{align*} 
4 
\begin{align*}.5^3\times .5=.063\end{align*} 
5 
\begin{align*}.5^4\times .5=.031\end{align*} 
6 
\begin{align*}.5^5\times .5=.016\end{align*} 
7 
\begin{align*}.5^6\times .5=.008\end{align*} 
8 
\begin{align*}.5^7\times .5=.004\end{align*} 
Vocabulary
Success and failure are associations given to outcomes based on their sequence. In the case of geometric distributions, a series of consecutive failures occur and then a single occurrence success occurs. Success should not necessarily have a positive connotation and failure does not need to have a negative one.
A trial is a single Binomial experiment with two possible outcomes whose results are independent in repeated experiments.
Guided Practice
Jordan is a student who hates to be called on in class. His teacher randomly calls on students and Jordan knows that on any given day he has a 10% chance of being called on. What is the probability that on the first day Jordan is called on? The 2nd day? The 5th day?
Answer:
Use the geometric distribution formula.
Number of Days  Probability 

1 
\begin{align*}.9^0\times .1=.1\end{align*} 
2 
\begin{align*}.9^1\times .1=.09\end{align*} 
5 
\begin{align*}.9^4\times .1=.059\end{align*} 
Practice
 Identify whether or not each situation has probabilities which are geometrically distributed:
 The number of plane flights someone makes before they are in a plane crash.
 The number of times your car will turn on before the battery dies.
 The number of times you win at a slot machine out of 5 plays.
 The number of gold medals a country wins in the Olympics.
 If the probability of winning a game is .4, then what is the probability of losing 5 games in a row and winning on the 6th game?
 If the probability of rolling a 4 on a fair die is \begin{align*}\frac{1}{6}\end{align*}
16 , what is the probability of rolling anything but fours for 10 rolls and then rolling a 4 on the 11th roll?  Suppose there is a game that is played until someone rolls a six, then the game ends. Create a probability distribution table for games ending in 8 or fewer rolls.
 If rain falls randomly with a 20% chance of rain on any given day, what is the probability that we suffer a drought for 8 days and then finally it rains on the 9th day?
 Suppose it rains like it does in question 5. What is the probability that we suffer a drought for 5 days and then get rain on any/all of the next 5 days?
 Suppose there is a jar of marbles. There are 10 red marbles and 2 green marbles. We randomly choose a marble and look at it. If it is red, we replace the marble into the jar and choose randomly again. If it is green, we stop. This process continues until we finally end with a green marble. What is the probability that the game ends just after the first draw?
 Using the same jar and game as question number 7, what is the probability that the game ends just after the second draw? What is the probability that the game ends just after the 3rd draw?
 Using the same jar and game as question number 7, what is the probability that the game takes more than 3 draws?
 Suppose you win blackjack 47% of the time. What is the probability that you lose 3 games in a row and then win on the 4th game?
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Here you’ll become familiar with another type of probability distribution called a geometric distribution. You’ll make connections with a binomial probability distribution and solve for probabilities of events.