1. 6.3 Day 2 Geometric Settings

2. Binomial vs Geometric
In a binomial setting, the number of trials (n) was fixed in advance, and the binomial random variable (X) counted the number of successes
Binary
Independent
Number
Success (probability) BINS
A geometric setting is similar. However, instead of having a fixed number of trials, we are counting HOW LONG IT TAKES until a particular outcome occurs.
Trials (counting the number of trials until the first success occurs)
Success
BITS

3. Geometric Random Variables
In a geometric setting, we can define a random variable (Y) to be the number of trials needed to get the first success. If we do this, Y is called a geometric random variable.
The probability distribution of Y is called a geometric distribution
To define a binomial random variable, we had 2 parameters
P (probability of success on each trial) and n (# trials)
For a geometric random variable, p is the only parameter

4. Is it geometric?
In the board game monopoly, one of the ways to get out of jail is to roll doubles (the same number on both dice). We want to know the probability that it will take us only one roll to do this.
Binary?
Independent?
Trials?
Success?

5. Is it geometric?
In the board game monopoly, one of the ways to get out of jail is to roll doubles (the same number on both dice). We want to know the probability that it will take us only one roll to do this.
Binary? YES—either roll doubles or don’t
Independent?
Trials?
Success?

6. Is it geometric?
In the board game monopoly, one of the ways to get out of jail is to roll doubles (the same number on both dice). We want to know the probability that it will take us only one roll to do this.
Binary? YES—either roll doubles or don’t
Independent? YES—not rolling doubles on the first trial doesn’t tell us anything about whether we will on the next turn/trial
Trials?
Success?

7. Is it geometric?
In the board game monopoly, one of the ways to get out of jail is to roll doubles (the same number on both dice). We want to know the probability that it will take us only one roll to do this.
Binary? YES—either roll doubles or don’t
Independent? YES—not rolling doubles on the first trial doesn’t tell us anything about whether we will on the next turn/trial
Trials? YES—we are counting how many trials until a success
Success?

8. Is it geometric?
In the board game monopoly, one of the ways to get out of jail is to roll doubles (the same number on both dice). We want to know the probability that it will take us only one roll to do this.
Binary? YES—either roll doubles or don’t
Independent? YES—not rolling doubles on the first trial doesn’t tell us anything about whether we will on the next turn/trial
Trials? YES—we are counting how many trials until a success
Success? YES—probability of rolling doubles doesn’t change

9. Geometric Probability
In our monopoly example, let’s calculate the probability of rolling doubles
To get a 1 on both dice: (1/6)(1/6)= 1/36
To get a 2 on both dice: (1/6)(1/6)= 1/36
To get a 3 on both dice: (1/6)(1/6)= 1/36
To get a 4 on both dice: (1/6)(1/6)= 1/36
To get a 5 on both dice: (1/6)(1/6)= 1/36
To get a 6 on both dice: (1/6)(1/6)= 1/36
So the total probability of rolling doubles is 6/36, or 1/6

10. Geometric Probability
So the probability of it taking one turn (trial) to roll doubles is 1/6= 6/36 =
The probability of it taking two turns (trials) to roll doubles is (1/6)(5/6)= 5/36 =
The 5/6 comes from the requirement that it did NOT occur on the first trial
The probability of it taking three turns is (1/6)(5/6)(5/6)= 25/216=
Because it had to NOT happen on both of the previous trials
Four turns: (1/6)(5/6)(5/6)(5/6)=
Notice the pattern: we multiply by an extra (5/6) each turn
5/6 is the probability of failure

11. Geometric Probability
This is exactly what we were just doing intuitively

12. Geometric Probability
𝑃 𝑌=1 = 1− − = 1 6 =
𝑃 𝑌=2 = 1− − = 5 36 =
𝑃 𝑌=3 = 1− − = = .1157
Etc.

13. On the Calculator
On any point, a certain tennis player has a 44% chance of making her first serve in. Assume that each serve is independent of the others. What is the probability that it takes her exactly 3 tries to make a first serve?
First check BITS
Binary?
Independent?
Trials?
Success?

14. On the Calculator
On any point, a certain tennis player has a 44% chance of making her first serve in. Assume that each serve is independent of the others. What is the probability that it takes her exactly 3 tries to make a first serve?
First check BITS
Binary? YES
Independent? YES
Trials? YES
Success? YES

15. On the Calculator
On any point, a certain tennis player has a 44% chance of making her first serve in. Assume that each serve is independent of the others. What is the probability that it takes her exactly 3 tries to make a first serve?
Now use the formula
𝑃 𝑌=3 = 1−.44 3− =.138
The probability that it takes her 3 tries to make a first serve is .138

16. Using our Calculator
Our calculator can directly calculate geometric probabilities as well
Geometpdf(p,k) calculates the probability that Y=k
Geometcdf(p,k) calculates the probability that Y≤k

17. Using our Calculator Let’s try the same problem with our calculator
For the tennis player that makes 44% of her first serves, use your calculator to calculate the probability that it takes her 3 tries to make a first serve
Pdf or cdf?

18. Using our Calculator Let’s try the same problem with our calculator
For the tennis player that makes 44% of her first serves, use your calculator to calculate the probability that it takes her 3 tries to make a first serve
Geometpdf(.44,3)
Answer: same as we got by hand

19. An Example
For the tennis player that makes 44% of her first serves, use your calculator to calculate the probability that it takes her 5 or fewer tries to make a first serve

20. An Example
For the tennis player that makes 44% of her first serves, use your calculator to calculate the probability that it takes her 5 or fewer tries to make a first serve
Geometcdf(.44,5) = .9449
Last one: what is the probability that it takes her more than 6 tries?

21. An Example
Last one: what is the probability that it takes her more than 6 tries?
1-geometcdf(.44,6)=
= .0308

22. AP Formula Sheet
The geometric probability formula is probably easier to use and more intuitive than the binomial probability formula
Geometric problems are also less common on the AP test (particularly free response)
For both of these reasons, the geometric probability formula is NOT on the AP formula sheet
But you do not need to memorize it

23. Getting Credit
If you DO memorize the geometric probability formula, plug p and k in to the formula for full credit
If not, clearly define your variable: “Y is a geometric random variable with p=.44. I therefore use geometpdf(.44,3) to find the probability that k=3
Or you can do it the way we did it at the beginning, where you treat it as a normal probability problem: “(.44)(.56)(.56)”

24. Shape of a Geometric Distribution
Notice that in both cases, the most likely result is that it only that it only takes 1 attempt
Always true—even when p=.01

25. Mean of a Geometric Distribution