Geometric Probability and Binomial Models in Engineering and Statistics

Agenda Warm Up Finish tournament? Geometric probability Study Hall Midterm review + Casino Project Make copies Upload Notes summary

Warm Up (AP) Engineers define reliability as the probability that a machine will perform its function. A certain model of aircraft engine is designed so that each engine has probability 0.999 of performing properly. Engineers test an SRS of 350 engines of this model. 1. Find the probability that all the engines perform properly. 2. Two engines fail the test. Are you convinced that this model of engine is unreliable? Justify with appropriate probability calculations. (Hint: Find P(x 348)

Yesterdays Exit Pass According to a 2000 study by the Bureau of Justice Statistics, approximately 2% of the nation s 72 million children had a parent behind bars nearly 1.5 million minors. Suppose that 100 children are randomly selected. Let X be the number of children who have an incarcerated parent. 1. Is X binomial? Justify. 2. Does your answer change if the children are sampled with replacement? Why? 3. Assume binomiality. Describe P(X=0) in a sentence, then calculate it. 4. What is the probability that 2 or more of the 100 children have a parent behind bars?

Liars Dice, rules Roll dice (hide from opponents). Loser decides who starts (may choose self) Players take turns. On your turn, you can . 1. Raise bet. Must say a higher bet. Bets are always at least bets. Bets are based on all dice on table. 1 s are wild (everything), unless someone bets 1 s. 2. Challenge . (can be said any time) All players reveal dice. Wrong person loses one die. 3. Exactly . All players reveal dice. Must be exactly that amount, no more, no less. If correct, all other players lose one die each. And repeat! Re-roll, start over. Winner = last player with dice.

Liars Dice Tournament Three Rounds 1stplace = 2pts 2ndplace = 1pt 3rd/4thplace = 0pts Schedule: Round 1 Lecture Binomial probability & Liar s Dice Mini-worksheet in groups Round 2 Round 3 Top 3 go to Championship Round in front of class Includes me 3 dice each

Liars Dice, math Use binomial to win! EXAMPLE: I m playing with 3 other people (16 dice total). I have four dice, and three of them are 6 s or 1 s. I bet seven 6 s What are the chances I m right? There are 12 more dice. I need four more 6 s or 1 s. Three or less .I lose. 2 binomcdf = 1 12 ( , ) 3 , 6 . 0 607

More examples There are 13 dice left among all players, including me. I have 5 dice left, so the other players have 8 dice. Three of my dice are 5 s or 1 s. 1. If I bet five 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 1 , 6 8049 . 0 2. If I bet seven 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 3 , 6 2587 . 0 3. If I bet ten 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 6 , 6 0026 . 0

Try it! worksheet

1. There are 15 dice left. You have 4 dice, with three 6 s. If you bet six 6 s , what is the probability that you are right? 2 binomcdf = 1 11 ( , ) 2 , 6 . 0 766 2. There are 10 dice left. You have 3 dice, with two 4 s. If you bet five 4 s , what is the probability that you are right? 2 binomcdf = 1 , 7 ( ) 2 , 6 4294 . 0 3. There are 18 dice left. You have 5 dice, with three 6 s. If you bet six 6 s , what is the probability that you are right? 2 binomcdf = 1 13 ( , ) 2 , 6 . 0 861 4. There are 16 dice left. But watch out, 1 s have been killed! You have 5 dice, with three 5 s. If you bet six 5 s , what is the probability that you are right? 1 binomcdf = 1 11 ( , ) 2 , 6 . 0 273

Liars Dice Tournament Three Rounds 1stplace = 2pts 2ndplace = 1pt 3rd/4thplace = 0pts Schedule: Round 1 Lecture Binomial probability & Liar s Dice Mini-worksheet in groups Round 2 Round 3 Top 3 go to Championship Round in front of class Includes me 3 dice each

Period 1 Round 2 1. Adeel, Yuki, Keren, Gabby E. 2. Christian, Elian, Megan, Gabriela H. 3. Julia, Harsh, Sydney, Jack 4. Jada, Nina, Michael, Daniel 5. Omar, Athena, Jonathan, Dipasha 6. Belinda, Koa, Andrew, Megha 7. Angie, Brian, Levani, Danny

Period 1 Round 3 1. Jonathan, Brian, Megha, Nina 2. Gabriela H., Michael, Gabby E. 3. Sydney, Levani, Megan, Omar 4. Andrew, Keren, Athena, Jack 5. Koa, Dipasha, Harsh, Angie 6. Daniel, Elian, Belinda, Jada 7. Yuki, Julia, Christian, Adeel

Period 3 Round 2 1. Adam, Rajesh, Laura, Dylan 2. Lukas, Maribel, Sangeeta, Ally 3. Eliseo, Diana, Hannah, Olivia 4. Hollie, Joven, Aidan, Eleny 5. Julian, Charmay, Drew, Fatima 6. Alfonso, Carolina, Tiarra, Mahek 7. Alicia, Ikram, Emily, Vanessa 8. Celia, Jade, Hassan, Danny

Period 3 Round 3 1. Fatima, Ally, Tiarra, Hannah 2. Eleny, Dylan, Drew, Sangeeta 3. Mahek, Olivia, Emily, Ikram 4. Hassan, Aidan, Laura, Carolina 5. Jade, Charmay, Maribel, Celia 6. Joven, Vanessa, Alicia, Hollie 7. Rajesh, Alfonso, Eliseo, Adam 8. Julian, Lukas, Diana

Geometric Probability

Geometric Notes 1 of 3 Properties of a geometric experiment: 1. B inary? Only 2 outcomes for each trial: success or failure. 2. I ndependent? Trials are independent. 3. N umber? Fixed number (n) of trials. 4. S uccess? Probability of success (p) is constant. These are geometric: a. Keep having kids until you get a daughter. b. Deal cards from a fair deck until you get a Heart. c. Roll a pair of dice until you get doubles. d. In basketball, attempt a 3-point shot until you make one. e. Keep playing a casino game until you win.

Window/Door Is it geometric? If not, explain why not.? 1. Shuffle a standard deck of playing cards. Then turn over one card at a time from the top of the deck, until you get an Ace. 2. Colligan is learning to shoot a bow and arrow. Colligan s instructor makes him keep shooting until he gets a bull s-eye. 3. The number of female births at Kaiser Permanente in Sacramento next week. 4. A student keeps taking a test until they get an A.

Window/Door 1. Shuffle a standard deck of playing cards. Then turn over one card at a time from the top of the deck, until you get an Ace. No, not constant probability of success. 2. Colligan is learning to shoot a bow and arrow. Colligan s instructor makes him keep shooting until he gets a bull s-eye. Probably not. No constant probability of success. 3. The number of female births at Kaiser Permanente in Sacramento next week. No. No. 4. A student keeps taking a test until they get an A.

Notes Geometric on Calculator 2 of 3 x is number of trials needed to succeed, with probability of success p. 2nd, Distr .. geometpdf (p,x) OR geometcdf (p,x) Precisely x trials needed At most x trials needed Examples: Roll 7 dice, get exactly three 3 s. Roll 7 dice, get at most three 3 s. binompdf (7, 1/6, 3) = 0.078 binomcdf (7, 1/6, 3) = 0.982 binomcdf (7, 1/6, 2) = 0.904 1 0.904 = 0.095 Roll 7 dice, get at least three 3 s.

Notes Geometric on Calculator 2 of 3 x is number of trials needed to succeed, with probability of success p. 2nd, Distr .. geometpdf (n,p,x) OR geometcdf (n,p,x) Precisely x trials needed At most x trials needed You roll a standard die. Keep rolling until you get a 3 . You get it on your 5th roll. You have to roll no more than 5 times. geometpdf (1/6, 5) = 0.0804 geometcdf (1/6, 5) = 0.598 geometcdf (1/6, 4) = 0.518 1 0.518 = 0.482 You have to roll at least 5 times.

Window/Door Find the probability of the following: 1. Flip a coin until you get Heads. You have to flip the coin 5 times. 2. Roll a die until you get a 6 . You have to roll the die 10 times. 3. Flip a coin until you get Heads. You have to flip the coin at least 5 times. 4. Roll a die until you get a 6 . You have to roll the die at least 10 times.

Window/Door ANSWERS (calculator) 1. Flip a coin until you get Heads. You have to flip the coin 5 times. 1 ( = geometpdf ) 5 , 2 03125 . 0 2. Roll a die until you get a 6 . You have to roll the die 10 times. 1 ( = geometpdf 10 , ) 0323 . 0 6 3. Flip a coin until you get Heads. You have to flip the coin at least 5 times. ) 4 , 2 1 geometcdf = 1 ( 0625 . 0 4. Roll a die until you get a 6 . You have to roll the die at least 10 times. ) 9 , 6 1 geometcdf = 1 ( 1938 . 0

Geometric Formula x 1 x number of trials needed p probability of success 1 ( ) p p Examples: Roll a standard die. Keeping rolling until you get a 3 . 1 1 5 1 1 ( ) ( ) You get it on your 5th roll. 6 6 You have to roll no more than 5 times. 1 ( ) 6 1 1 1 1 1 1 1 1 1 + + + + 5 1 4 1 3 1 2 1 1 1 1 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 6 6 6 6 6 6 6 6 6

Example: FRQ 2011 #3 An airline claims that there is a 0.10 probability that a coach-class ticket holder who flies frequently will be upgraded to first class on any flight. This outcome is independent from flight to flight. Sam is a frequent flier who always purchases coach-class tickets. a. What is the probability that Sam s first upgrade will occur after the third flight? 0.10 probability (p) 3 trials until success (x) geometcdf = 1 . 0 ( 10 ) 3 , . 0 729 b. What is the probability that Sam will be upgraded exactly 2 times in his next 20 flights? Sam will take 104 flights next year. Would you be surprised if Sam receives more than 20 upgrades to first class during the year? Justify your answer. c.

Notes Mean and s.d. of Geometric 3 of 3 If you roll a die, and you re trying to get a 3 , how many times will you probably have to roll before you get a 3 . How much variability? 1 = 1 p = p p Example: If you roll a die 15 times before getting a 3 , do you conclude that the die is weighted? 1 = = 1 1 6 6 = = . 5 477 1 1 6 6 15 6 . 1 = 64 = 1 9495 . 0 0505 . 0 . 5 477

Example: FRQ 2011 #3 An airline claims that there is a 0.10 probability that a coach- class ticket holder who flies frequently will be upgraded to first class on any flight. This outcome is independent from flight to flight. Sam is a frequent flier who always purchases coach-class tickets. a. What is the probability that Sam s first upgrade will occur after the third flight? geometcdf = 1 . 0 ( 10 ) 3 , . 0 729 1 1 . 0 10 = = 10 = = . 9 49 . 0 10 . 0 10 3 10 = . 0 74 = 1 2296 . 0 7704 . 0 . 9 49

Try it! 1. To start her mower, Rita has to pull a cord and hope for some luck. On any particular pull, the mower has a 20% chance of starting. Find the probability that it takes her a. exactly 3 pulls to start her mower. b. At most 10 pulls to start her mower. c. At least 10 pulls to start her mower. 2. Martin decided to keep playing Roulette over and over, placing a $1 bet on his lucky number 15. On any spin, there s a 1-in-38 chance that the ball will land in the 15 slot. a. How many spins do you expect it to take until Martin wins? b. Would it surprise you if Martin won in 3 or fewer spins? Why or why not? Justify with probability calculations.

Try it! ANSWERS 1. To start her mower, Rita has to pull a cord and hope for some luck. On any particular pull, the mower has a 20% chance of starting. Find the probability that it takes her a. exactly 3 pulls to start her mower. = . 0 ( 20 ) 3 , . 0 128 geometpdf b. At most 10 pulls to start her mower. = . 0 ( 20 10 , ) . 0 893 geometcdf c. At least 10 pulls to start her mower. 1 geometcdf = . 0 ( 20 ) 9 , . 0 134

Try it! ANSWERS 2. Martin decided to keep playing Roulette over and over, placing a $1 bet on his lucky number 15. On any spin, there s a 1-in-38 chance that the ball will land in the 15 slot. a. How many spins do you expect it to take until Martin wins? 1 = = 38 1 38 b. Would it surprise you if Martin won in 3 or fewer spins? Why or why not? Justify with probability calculations. 1 = ( ) 3 , . 0 077 geometcdf 38

Study Hall 1. Study for your midterm. If you want to study for your midterm next week: Start the review packet. It s 38 multiple-choice and 3 free-response questions, organized chronologically by Unit, with answers and explanations on the last page. You re welcome. Look through the Notes Summary. It s every Notes slide from all our lessons. 2. Work on your Casino Project. Calculations are due next Wednesday unless you negotiate with me. Doing a confidence interval? Don t forget your time-lapse!

Exit Pass A psychologist studied the number of puzzles that subjects were able to solve in a five-minute period while listening to soothing music. This is the probability distribution for X number of puzzles. Value of X 1 2 3 4 Prob. 0.2 0.4 0.3 0.1 1. What is the expected number of solved puzzles? 2. Suppose that three randomly selected subjects solve puzzles for five minutes each. What is the expected value of the total number of puzzles solved by the three subjects?