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4A: Probability Concepts and Binomial Probability Distributions

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

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  1. 4A: Probability Concepts and Binomial Probability Distributions Probability Concepts & Binomial Distributions

  2. 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

  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 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) Probability Concepts & Binomial Distributions

  5. 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

  6. Rules for Probabilities 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, 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

  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) 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 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) Probability Concepts & Binomial Distributions

  11. 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

  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”: Probability Concepts & Binomial Distributions

  13. 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

  14. 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

  15. 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

  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) Probability Concepts & Binomial Distributions

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

  19. The Cumulative Mass Function for X~b(4, 0.75) Probability Concepts & Binomial Distributions

  20. 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

  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). 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) Probability Concepts & Binomial Distributions

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