1. 4A: Probability Concepts and Binomial Probability Distributions
12/31/2018
4A: Probability Concepts and Binomial Probability Distributions
12/31/2018
Probability Concepts & Binomial Distributions
Biostat

2. Probability Concepts & Binomial Distributions
Definitions
Random variable  a numerical quantity that takes on different values depending on chance
Population  the set of all possible values for a random variable
Event  an outcome or set of outcomes for a random variable
Probability  the proportion of times an event occurs in the population; (long-run) expected proportion
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Probability Concepts & Binomial Distributions

3. Probability (Definition #1)
Probability is its relative frequency of the event in the population.
Example:
Let A  selecting a female at random from an HIV+ population
There are 600 people in the population.
There are 159 females.
Therefore, Pr(A) = 159 ÷ 600 = 0.265
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Probability Concepts & Binomial Distributions

4. Probability (Definition #2)
Probability is the long run proportion when the process in repeated again and again under the same conditions.
Select 100 individuals at random
24 are female
Pr(A)  24 ÷ 100 = 0.24
This is only an estimate (unless n is very very big)
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Probability Concepts & Binomial Distributions

5. Probability (Definition #3)
Probability is a quantifiable level of belief between 0 and 1
Probability
Verbal expression
0.00
Never
0.05
Seldom
0.20
Infrequent
0.50
As often as not
0.80
Very frequent
0.95
Highly likely
1.00
Always
Example: I believe a quarter of population is male. Therefore, in selecting individuals at random: Pr(male) ≈ 0.25
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Probability Concepts & Binomial Distributions

6. Rules for Probabilities
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Probability Concepts & Binomial Distributions

7. Types of Random Variables
Discrete have a finite set of possible outcomes,
e.g. number of females in a sample of size n (0, 1, 2, …, n)
We cover binomial random variables
Continuous have a continuum of possible outcomes
e.g., average body weight (lbs) in a sample (160, 160.5, , , …)
We cover Normal random variables
There are other random variable families, but only binomial (this lecture) and Normal (next lecture) families will be covered.
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Probability Concepts & Binomial Distributions

8. Binomial random variables
Most popular type of discrete random variable
Bernoulli trial  random event characterized by “success” or “failure”
Examples
Coin flip (heads or tails)
Survival (yes or no)
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Probability Concepts & Binomial Distributions

9. Binomial random variables (cont.)
Binomial random variable  random number of successes in n independent Bernoulli trials
A family of distributions identified by two parameters
n  number of trials
p  probability of success for each trial
Notation: X~b(n,p)
X  random variable
~  “distributed as”
b(n, p)  binomial RV with parameters n and p
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Probability Concepts & Binomial Distributions

10. “Four patients” example
A treatment is successful 75% of time
We treat 4 patients
X  random number of successes, which varies  0, 1, 2, 3, or 4 depending on binomial distribution X~b(4, 0.75)
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Probability Concepts & Binomial Distributions

11. The probability of i successes is …
The Binomial Formula
The probability of i successes is …
Where
nCi = the binomial coefficient (next slide)
p = probability of success for each trial
q = probability of failure = 1 – p
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Probability Concepts & Binomial Distributions

12. Binomial Coefficient (“Choose Function”)
where
!  the factorial function: x! = x  (x – 1)  (x – 2)  …  1
Example: 4! = 4  3  2  1 = 24 By definition 1! = 1 and 0! = 1
nCi  the number of ways to choose i items out of n
Example: “4 choose 2”:
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Probability Concepts & Binomial Distributions

13. The “Four Patients” Illustrative Example
n = 4 and p = 0.75 (so q = = 0.25)
Question: What is probability of 0 successes?  i = 0
Pr(X = 0) =nCi pi qn–i = 4C0 · · 0.254–0 = 1 · · =
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Probability Concepts & Binomial Distributions

14. Probability Concepts & Binomial Distributions
X~b(4,0.75), continued
Pr(X = 1) = 4C1 · · –1
= 4 · · =
Pr(X = 2) = 4C2 · · –2
= 6 · · =
(Do not demonstrate all calculations. Students should prove to themselves they derive and interpret these values.)
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Probability Concepts & Binomial Distributions

15. Probability Concepts & Binomial Distributions
X~b(4, 0.75) continued
Pr(X = 3) = 4C3 · · –3
= 4 · · 0.25 =
Pr(X = 4) = 4C4 · · –4
= 1 · · 1 =
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Probability Concepts & Binomial Distributions

16. The Probability Mass Function for X~b(4, 0.75)
Probability table for X~b(4,.75)
Probability curve for X~b(4,.75)
Successes
Probability
0.0039 1
0.0469 2
0.2109 3
0.4210 4
0.3164
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Probability Concepts & Binomial Distributions

17. Area Under The Curve (AUC)
The area under the curve (AUC) = probability!
Get it?
Pr(X = 2) = .2109
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Probability Concepts & Binomial Distributions

18. Cumulative Probability (left tail)
Cumulative probability = Pr(X  i) = probability less than or equal to i
Illustrative example: X~b(4, .75)
Pr(X  0) = Pr(X = 0) = .0039
Pr(X  1) = Pr(X  0) + Pr(X = 1) = =
Pr(X  2) = Pr(X  1) + Pr(X = 2) = =
Pr(X  3) = Pr(X  2) + Pr(X = 3) = =
Pr(X  4) = Pr(X  3) + Pr(X = 4) = =
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Probability Concepts & Binomial Distributions

19. The Cumulative Mass Function for X~b(4, 0.75)
Probability function
Cumulative probability
Pr(X  0)
0.0039
Pr(X  1)
0.0469
0.0508
Pr(X  2)
0.2109
0.2617
Pr(X  3)
0.4210
0.6836
Pr(X  4)
0.3164
1.0000
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Probability Concepts & Binomial Distributions

20. Cumulative Probability
Area under left tail = cumulative probability
Area under shaded bars in left tail sums to : Pr(X  2) =
Area under “curve” = probability
Bring it on!
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Probability Concepts & Binomial Distributions

21. Reasoning with Probabilities
We use probability model to reasoning about uncertainty & chance.
I hypothesize p = 0.75, but observe only 2 successes. Should I doubt my hypothesis?
ANS: No. When p = 0.75, you’ll see 2 or fewer successes 25% of the time (not that unusual).
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Probability Concepts & Binomial Distributions

22. StaTable Probability Calculator
Three versions
Java (browser)
Windows
Palm
Calculates probabilities for many pmfs and pdfs
Example (right) is for a X~b(4,0.75) when x = 2
No of successes x
Pr(X = x)
Pr(X ≤ x)
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Probability Concepts & Binomial Distributions