1. Binomial Distribution And general discrete probability distributions...

2. Random Variable A random variable assigns a number to a chance outcome or chance event. The definition of the random variable is denoted by uppercase letters at the end of alphabet, such as W, X, Y, Z. The possible values of the random variable are denoted by corresponding lowercase letters w, x, y, z.

3. Examples: Discrete random variables W = number of beers randomly selected student drank last night, w = 0, 1, 2, … X = number of aspirin randomly selected student took this morning, x = 0, 1, 2,... Y = number of children in a family with children, y = 1, 2, 3,...

4. Discrete Probability Distribution A discrete probability distribution specifies: –the possible values of the random variable, and –the probability that each outcome will occur

5. Example: Discrete probability distribution Let X = number of natural brothers PSU students have. X = 0, 1, 2,... P(X=0) = 0.41 P(X=1) = 0.45 P(X=2) = 0.11 P(X=3) = 0.03

6. Example: Graphically …

7. Example: Discrete probability distribution Let X = the number selected when a student picks a number between 0 and 9. If students pick number randomly, then the probability of picking any number is 0.10. That is, P(X = 0) = … = P(X = 9) = 0.10

8. Example: Binomial random variable Student told 3 statements -- 2 true, 1 false. Student tries to identify false statement. Student does this 3 different times. Let X = the number of times student correctly identifies the false statement. Then, X = 0, 1, 2 or 3.

9. independent trials: a student’s success or failure on one try doesn’t affect success or failure on another try Binomial Random Variable A special kind of discrete random variable having the following four characteristics: 2 possible outcomes denoted “success” or “failure”: student picks either false statement or true statement. p = P(“success”) is same for each trial: if just guessing, a student has probability of 1/3 of picking false statement. n identical “trials”: student tries to guess 3 times

10. Is X binomial? Probability student smokes pot regularly is 0.25. College administrator surveys students until finds one who smokes pot. Let X = number of students surveyed.

11. Is X binomial? Unknown to quality control inspector, crate of 50 light bulbs contain 3 defective bulbs. QC inspector randomly selects 5 bulbs “without replacement”. Let X = number of defective bulbs in inspector’s sample.

12. Is X binomial? Unknown to us, the probability an American thinks Clinton should have been removed from office is 0.29. Gallup poll surveys 960 Americans. Let X = number of Americans in sample who think Clinton should have been removed from office.

13. Is X binomial? Students pick one number between 0 and 9. Let X = number of students who pick the number “7”

14. Example: Binomial r.v. Let 3 students pick. Let Y = #7 and N = not #7

15. Binomial Probability Distribution P(X = x) = (# of ways x occurs) × p x × (1-p) n-x = n!/[x!(n-x)!] × p x × (1-p) n-x Where “n-factorial” is defined as n!= n (n-1) (n-2) … 1 and 0! = 1

16. Examples: n! 5! = 5 × 4 × 3 × 2 × 1 = 120 4! = 4 × 3 × 2 × 1 = 24 3! = 3 × 2 × 1 = 6 2! = 2 × 1 = 2 1! = 1

17. Example: Binomial Formula Guessing game. Let n = 3 and p = 0.33. Then: P(X = x) = n!/[x!(n-x)!] × p x × (1-p) n-x P(X = 0) = 3!/[0!(3-0)!] × 0.33 0 × (0.67) 3-0 = 6/(1×6) × 1 × 0.67 3 = 0.30 P(X = 1) = 3!/[1!(3-1)!] × 0.33 1 × (0.67) 3-1 = 3 × 0.33 × 0.67 2 = 0.44

18. Example (continued) P(X = 2) = 3!/[2!(3-2)!] × 0.33 2 × (0.67) 3-2 = 3 × 0.1089 × 0.67 = 0.22 P(X = 3) = 3!/[3!(3-3)!] × 0.33 3 × (0.67) 3-3 = 1 × 0.037 × 1 = 0.04 Note: 0.30 + 0.44 + 0.22 + 0.04 = 1

19. Using binomial probabilities to draw a conclusion If students did just randomly guess which statement was false, we’d expect the random variable X to follow a binomial distribution? Can we conclude that students did not guess the false statements randomly?

20. Using binomial probabilities to draw a conclusion If students do indeed pick a number between 0 and 9 randomly, how likely is it that we would observe the sample we did? Can we conclude that students do not pick numbers randomly?

21. Using binomial probabilities to draw a conclusion Could the space shuttle Challenger disaster of January 28, 1986 have been better predicted? And therefore prevented?

22. Moral Probability calculations are used daily to draw conclusions and make important decisions. Calculated probabilities are accurate only if the assumptions made are indeed correct. Always check to see if your assumptions are reasonable.