Probability Class 32 1

2 Two-Way Table: “Hypothetical Hundred Thousand” Example 7.25 Teens and Gambling Sample of ninth-grade teens: 49.1% boys, 50.9% girls. Results: 22.9% of boys and 4.5% of girls admitted they gambled at least once a week during previous year. Start with hypothetical 100,000 teens … (.491)(100,000) = 49,100 boys and thus 50,900 girls Of the 49,100 boys, (.229)(49,100) = 11,244 would be weekly gamblers. Of the 50,900 girls, (.045)(50,900) = 2,291 would be weekly gamblers.

3 Example 7.25 Teens and Gambling P(boy and gambler) = 11,244/100,000 = P(boy | gambler) = 11,244/13,535 = P(gambler) = 13,535/100,000 =

4 Example 7.26 Alicia’s Possible Fates P(Alicia has D and has a positive test) = P(test is positive) = = P(Alicia has D | positive test) =.00095/.0509 =.019

5 Example 7.28 Teens and Gambling P(boy and gambler) = (.491)(.229) =.1124 P(girl and not gambler) = (.509)(.955) =.4861 P(gambler) = =.1353 P(boy | gambler) =.1124/.1353 =.8307

6 Tree Diagrams Step 1: Determine first random circumstance in sequence, and create first set of branches for possible outcomes. Create one branch for each outcome, write probability on branch. Step 2: Determine next random circumstance and append branches for possible outcomes to each branch in step 1. Write associated conditional probabilities on branches. Step 3: Continue this process for as many steps as necessary. Step 4: To determine the probability of following any particular sequence of branches, multiply the probabilities on those branches. This is an application of Rule 3a. Step 5: To determine the probability of any collection of sequences of branches, add the individual probabilities for those sequences, as found in step 4. This is an application of Rule 2b.

7 7.6 Using Simulation to Estimate Probabilities Some probabilities so difficult or time- consuming to calculate – easier to simulate. If you simulate the random circumstance n times and the outcome of interest occurs in x out of those n times, then the estimated probability for the outcome of interest is x/n.

How to do simulation Question: Toss a coin 10 times. What is the probability of a run of at least 3 consecutive heads or 3 consecutive tails? Step 1: Determine if it is a probability model –Each toss has probabilities 0.5 for a head and 0.5 for a tail –Tosses are independent of each other

How to do simulation…cont’d Step 2: Assign digits to represent outcomes –The random number table has probability of 0.1 of being any one of 0,1,2,3,4,5,6,7,8,9 –Successive digits in the table are independent –One digit simulates one toss of the coin –Odd digits represent heads (0.5) ; even digits represent tails (0.5). Step 3: Simulate many repetitions –10 digits represent 10 tosses –Example: –represent H H T T H H H T H T Step 4: Did 25 repetitions….Estimate the probability –Example: if 23 out of 25 repetitions have a run of 3 consecutive heads or 3 consecutive tails –The estimated probability = 23/25

Assigning Digits for Simulation 1.Choose a person at random from a group which 50% are employed –One digit simulates one person: Odd number – unemployed Even number – employed –OR: 0, 1, 2, 3, 4 = employed 5, 6, 7, 8, 9 = unemployed

Assigning Digits for Simulation…cont’d 1.Choose a person at random from a group which 70% are employed –HOW?? –One digit simulates one person: 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 = unemployed

Assigning Digits for Simulation…cont’d 2. Choose a person at random from a group of which 73% are employed. HOW?? –Two digits simulates one person: 00, 01, 02,……….72 = employed 73, 74, 75………..99 = unemployed

Assigning Digits for Simulation…cont’d 3. Choose a person at random from a group of which 50% are employed, 20% are unemployed, 30% are not in the labor force HOW?? –One digit simulates one person: 0, 1, 2, 3, 4 = employed 5, 6 = unemployed 7, 8, 9 = not in the labor force

Quick Check 1 An opinion poll selects adult Americans at random and asks them, “Which political party, Democratic or Republican, do you think is better able to manage the economy?” Explain carefully how you would assign digits from the random number table to simulate the response of one person in each of the following situations. 1.Of all adult Americans, 50% would chose the Democrats and 50% the Republicans 2. Of all adult Americans, 60% would chose the Democrats and 40% the Republicans 3.Of all adult Americans, 55% would chose the Democrats and 45% the Republicans 4.Of all adult Americans, 40% would chose the Democrats and 40% the Republicans and 20% are undecided 5.Of all adult Americans, 50% would chose the Democrats and 38% the Republicans and 12% are undecided

Quick Check 2 Shooting Baskets Mr. Myer shot free throws at a 70% this season. How likely is it that he would make 7 shots in a row out of 10 shots? (Simulate 15 repetitions) Mr. Jameson shoots free throws at a 54% rate. How often would he make 7 in a row out of 10 shots? (Simulate 15 repetitions)

Use Tree diagram to do simulation Question: Morris’s kidneys have failed and he is awaiting a kidney transplant. His doctor gives him this information for patients in his condition: 90% survive the transplant operation, and 10% die. The transplant succeeds in 60% of those who survive, and the other 40% must return to kidney dialysis. The proportions who survive for at least five years are 70% for those with a new kidney and 50% for those who return to dialysis. Morris wants to know the probability that he will survive for at least 5 years.

Use Tree diagram to do simulation Step 1: Construct a tree diagram Surviv e New Kidney Surviv e Die Dialysis Di e

Use Tree diagram to do simulation Step 2: Assign digits to outcome Stage 1: 0 = die 1,2,3,4,5,6,7,8,9 = survive Stage 2: 0, 1, 2, 3, 4, 5 = transplant succeeds 6, 7, 8, 9 = return to dialysis Stage 3 with new kidney: 0, 1, 2, 3, 4, 5, 6 = survive for 5 years 7, 8, 9 = die Stage 3 with dialysis: 0, 1, 2, 3, 4 = survive for 5 years 5, 6, 7, 8, 9 = die

Use Tree diagram to do simulation Step 3: Use random number table to do simulations of several repetitions to determine the estimated probability of “survive five years”, each arranged vertically For example: random number: Repetition1Repetition 2Repetition 3Repetition 4 Stage 11 -> Survive3 -> Survive7 -> Survive4 -> Survive Stage 27 -> return to dialysis 8 -> return to dialysis 5 -> New Kidney2 -> New Kidney Stage 31 -> Survive2 -> Survive8 -> Die5 -> Survive Morris survives five years in 3 of the 4 repetitions. The estimated probability = 3/ 4

Class Task 3 : Use Tree diagram to do simulation Question: Erica is about to graduate from TRMC. Her guidance counselor gives her this information: 80% of graduates go to college and 20% go to work. Of 80% graduates go to college 70% go to a 4-year college and the other 30% go to 2-year community college. 60% of those who go to community college will graduate with an associate degree and the other 40% will drop out and go to work. 50% of those who go to 4-year college will graduate with a bachelor degree and the rest drop out and go to work. Erica wants to know the probability that she will graduate with either an Associate degree of Bachelor degree.

Homework Assignment: Chapter 7 – Exercise 7.68, 7.69 and 7.70 Reading: Chapter 7 – p