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4A: Probability Concepts and Binomial Probability Distributions

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### 4A: Probability Concepts and Binomial Probability Distributions

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

Probability Concepts & Binomial Distributions

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

Probability Concepts & Binomial Distributions

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)

Probability Concepts & Binomial Distributions

Probability (Definition #3)

Probability is a quantifiable level of belief between 0 and 1

Example: I believe a quarter of population is male. Therefore, in selecting individuals at random: Pr(male) ≈ 0.25

Probability Concepts & Binomial Distributions

Rules for Probabilities

Probability Concepts & Binomial Distributions

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, 160.75, 160.825, …)
- We cover Normal random variables

There are other random variable families, but only binomial (this lecture) and Normal (next lecture) families will be covered.

Probability Concepts & Binomial Distributions

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)

Probability Concepts & Binomial Distributions

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

Probability Concepts & Binomial Distributions

“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)

Probability Concepts & Binomial Distributions

The probability of i successes is …

The Binomial FormulaWhere

nCi= the binomial coefficient (next slide)

p = probability of success for each trial

q = probability of failure =1 – p

Probability Concepts & Binomial Distributions

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”:

Probability Concepts & Binomial Distributions

The “Four Patients” Illustrative Example

- n = 4 and p = 0.75 (so q = 1 - 0.75 = 0.25)
- Question: What is probability of 0 successes? i = 0
- Pr(X = 0) =nCi pi qn–i = 4C0 · 0.750 · 0.254–0= 1 · 1 · 0.0039 = 0.0039

Probability Concepts & Binomial Distributions

X~b(4,0.75), continued

Pr(X = 1) = 4C1· 0.751 · 0.254–1

= 4 · 0.75 · 0.0156

= 0.0469

Pr(X = 2) = 4C2· 0.752 · 0.254–2

= 6 · 0.5625 · 0.0625

= 0.2106

(Do not demonstrate all calculations. Students should prove to themselves they derive and interpret these values.)

Probability Concepts & Binomial Distributions

X~b(4, 0.75) continued

Pr(X = 3) = 4C3· 0.753 · 0.254–3

= 4 · 0.4219 · 0.25

= 0.4219

Pr(X = 4) = 4C4· 0.754 · 0.254–4

= 1 · 0.3164 · 1

= 0.3164

Probability Concepts & Binomial Distributions

The Probability Mass Function for X~b(4, 0.75)

Probability table for X~b(4,.75)

Probability curve for X~b(4,.75)

Probability Concepts & Binomial Distributions

Pr(X = 2) = .2109

Area Under The Curve (AUC)The area under the curve (AUC) = probability!

Probability Concepts & Binomial Distributions

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) = .0039 + .0469 = 0.0508
- Pr(X 2) = Pr(X 1) + Pr(X = 2) = .0508 + .2109 = 0.2617
- Pr(X 3) = Pr(X 2) + Pr(X = 3) = .2617 + .4219 = 0.6836
- Pr(X 4) = Pr(X 3) + Pr(X = 4) = .6836 + .3164 = 1.0000

Probability Concepts & Binomial Distributions

The Cumulative Mass Function for X~b(4, 0.75)

Probability Concepts & Binomial Distributions

Cumulative Probability

Area under left tail = cumulative probability

Area under shaded bars in left tail sums to 0.2617:Pr(X 2) = 0.2617

Area under “curve” = probability

Probability Concepts & Binomial Distributions

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).

Probability Concepts & Binomial Distributions

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)

Probability Concepts & Binomial Distributions

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