Advanced Placement Statistics Section 8 1 Binomial Distributions

Advanced Placement Statistics Section 8. 1: Binomial Distributions (Part I) EQ: How do you solve probability questions in a binomial setting?

Terms to Recall/Know: Random Variable X discrete number 0 to n Binary – 2 possible outcomes Examples: ON or OFF MALE or FEMALE HIT or MISS DOES NOT IMPLY 50%/50% PROBABLITY OF OUTCOMES

Binomial Random Variable X # of successful outcomes Binomial Probability Distribution B(n, p) n # of total outcomes p probability of success for any 1 observation q probability of failure for any 1 observation

parameter population Four Properties of a Binomial Experiment: 1. fixed number of trials n = # of [in context] 2. each trial results in a success or failure (disjoint) S= [define success] F= [define failure]

3. outcomes of trials are independent 4. probability of each success is the same from trial to trial Recall: Complements If S is success, and F is failure, then P(F) = 1 – P(S).

Read Ex 8. 1, p. 514 Does this meet the 4 properties of a binomial distribution? 1) Fixed number of trials? 2) Success/Failure? 3) Independence? n= 5 children S = having type O blood F = not having type O blood Blood type is independent from one person to another 4) Probability of success the same? Probability of p = 0. 25 and q = 0. 75 remains the same from trial to trial.

Read Ex 8. 2, p. 514 Does this meet the 4 properties of a binomial distribution? 1) Fixed number of trials? 2) Success/Failure? 3) Independence? NO, IT’S WITHOUT 4) Probability of success the same? REPLACEMENT!! Therefore this distribution cannot be written B(n, p)

In class Assignment: p. 516 #1 - 6 #1 NO; n not a fixed value #2 YES; define all properties in context #3 YES; define all properties in context #4 NO; n not a fixed value #5 NO; probability of success likely not to remain the same b/c remediation given for incorrect responses #6 YES; define all properties in context

Handout WS Examples Section 8. 1 Complete Ex 1 Ex. 1 The Heart Association claims that only 10% of American adults over 30 can pass the President’s Physical Fitness Commission minimum requirements. Suppose 4 adults are randomly selected and each is given the fitness test. a) What is the observation of interest? The observation of interest is the result (pass or fail) of the fitness test. b) Justify in context of the problem that this setting meets the properties of a binomial distribution. • fixed number of trials; n = 4 adults randomly selected to take the fitness test • success = adult passes test, failure = adult fails the test • Each adult’s test result is independent of another adult’s test result. • Probability of success and failure remains the same from trial to trial. P(passes test) = 0. 1 P(fails test) = 0. 9

Day 45 Agenda: Quiz 7. 1 & 7. 2 --- 30 minutes

SSSS FFSS FSFS FSSF SFFS SFSF SSFF FSSS SFSS SSFS SSSF SSSS (. 1)(. 1) = 0. 0001 SFFF FSFF FFSF FFFS FFFF Intersection; they must happen simultaneously FFFF (. 9)(. 9) = 0. 6561 SSSF SSFF SFFF X 0 1 2 3 4 P(X) . 6561 . 2916 . 0486 . 0036 . 0001 (. 1)(. 9)(4) =. 0036 (. 1)(. 9)(6) =. 0486 (. 1)(. 9)(4) =. 2916 . 6561+. 2916+. 0486+. 0036+. 0001 = 1

RECALL: Formulas to Know: n! “n factorial” n(n - 1)(n – 2)…(n – (n -1)) n(n - 1)(n – 2)…(1) Ex. 5 ! = (5)(4)(3)(2)(1) = 120 Combinations also called the “binomial coefficient”

Binomial Probability Find this on your formula sheet! Go back to WS Ex 1. Find f) and g): X = number of adults who pass a fitness test out of 4 randomly selected adults taking the test f) P(X = 3) = g) P(X = 2) =

RECALL: CALCULATOR FUNCTIONS: binomialpdf (n, p, x) For one specific value Ex. Find the probability of 5 successes in 8 trials where the probability of success is 40%. 0. 1239 P(X = 5) = ______ binomialpdf (8, 0. 4, 5)

Cumulative Probability Density Function binomialcdf (n, p, x) For cumulative values Ex. Find the probability of less than 5 successes in 8 trials where the probability of success is 40%. P(X < 5) = _____ means P(X < 4) =0. 8263 ____ 0. 8263 P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = _____ binomialcdf (8, 0. 4, 4)

Ex. Find the probability of at least 5 successes in 8 trials where the probability of success is 40%. 0. 1736 1 - binomialcdf (8, 0. 4, 4) Ex. Find the probability of more than 5 successes in 8 trials where the probability of success is 40%. 0. 0498 1 - binomialcdf (8, 0. 4, 5)

Go back to Handout WS Examples Section 8. 1 Complete Ex 2 Ex. 2 Periodically the FTC monitors the pricing accuracy of electric checkout scanners at stores to ensure consumers are charged the correct price at checkouts. A 2004 study of over 100, 000 items found that one of every 30 items is priced incorrectly by scanners. Suppose the FTC randomly selected five items at a retail store and checks the accuracy of the scanner price of each. a) Determine if this experiment meets the criteria of a binomial experiment. • fixed number of trials; n = 5 items scanned • success = item accurately scanned, failure = item inaccurately scanned • Result of one item scanned is independent of the result of another item scanned. • Probability of success and failure remains the same from trial to trial. P(item accurately scanned) = 29/30 P(item inaccurately scanned) = 1/30

a) Define the random variable X then create a probability distribution table. X = number of items accurately scanned out of 5 randomly selected items scanned at a retail store X 0 1 2 P(X) 4. 12 x 10 -8 5. 97 x 10 -6. 0003 3 4 . 01004. 1455 5. 8441 RECALL: But how can you use lists to calculate these values? X L 1 binomialpdf(5, 29/30, L 1) L 2

a) Define the random variable X then create a probability distribution table. X = number of items accurately scanned out of 5 randomly selected items scanned at a retail store X 0 1 2 3 P(X) 4. 12 x 10 -8 5. 97 x 10 -6. 0003 4 5 . 01004. 1455 . 8441 a) Create a cumulative probability distribution table. X F(X) 0 1 2 4. 12 x 10 -8 6. 01 x 10 -6 3. 52 x 10 -4 3 4 . 0104 . 1559 binomialcdf(5, 29/30, L 1) 5 1

d) What is the probability that at most 2 items are priced incorrectly? X>3 = P(____) . 9997 means at least 3 are priced correctly e) What is the probability that at least one of the 5 items is priced incorrectly? means at most 4 are priced correctly X < 4 0. 1559 P( _____) = 1 – P(X = 5)= 1 – 0. 8441 = 0. 1559

ASSIGNMENT: p. 523 - 524 #13 - 16