4a probability concepts and binomial probability distributions
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4A: Probability Concepts and Binomial Probability 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

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4a probability concepts and binomial probability distributions

4A: Probability Concepts and Binomial Probability Distributions

Probability Concepts & Binomial Distributions

definitions
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 (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 (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 (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
Rules for Probabilities

Probability Concepts & Binomial Distributions

types of random variables
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
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 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
“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 binomial formula

The probability of i successes is …

The Binomial Formula

Where

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
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
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
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 continued1
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
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

area under the curve auc

Get it?

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

Probability Concepts & Binomial Distributions

cumulative probability

Bring it on!

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
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
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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