### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# ProbStats MATH 1530

ETSU

GPA 3.5

### View Full Document

## 14

## 0

## Popular in Course

## Popular in Mathematics (M)

This 13 page Class Notes was uploaded by Ms. Ismael Spinka on Sunday October 11, 2015. The Class Notes belongs to MATH 1530 at East Tennessee State University taught by Edith Seier in Fall. Since its upload, it has received 14 views. For similar materials see /class/221402/math-1530-east-tennessee-state-university in Mathematics (M) at East Tennessee State University.

## Similar to MATH 1530 at ETSU

## Popular in Mathematics (M)

## Reviews for ProbStats

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/11/15

ROBE MdTh 1530 005 009 amp 014 Fdll 2003 LecTure NoTes for The Binomial DisTribuTions pdrT of ch 17 Consider d ciTy wiTh Thousands of cors in iT Suppose Tth 30 of The cars in The ciTy ore red Now suppose you pick one cor dT rdndom from The ciTy thT is The probdbiIiTy Tth iT is 0 red cor Pred PnoT red Suppose I pick Two cors dT rdndom The cors ore independenT of one dnoTher SepdrdTer PfirsT cor is red Psecond cor is red For Two or more independenT evenTs probdbiIiTies muITipIy TogeTher PboTh cors ore red PfirsT is red and second is red PfirsT is red X Psecond is red 2 X Z PboTh ore noT red x PfirsT is red and second is noT x PfirsT is noT and second is red x Pone red in The Two corsdmple 22 Now The ided one red cor ouT of The Two chosen is on EvenT wiTh Two possible ouTcomes in iT firsT is red and second is noT or firsT is noT and second is red ProbdbiIiTies musT be colculdTed dccordingly Pone red in The Two cor sdmple P firsT is red and second is noT or firsT is noT and second is red Because these two are disjoint we can just add PfirsT is red and second is noT PfirsT is noT and second is red 2 2 Observe pick Two cors dT rdndom consisTs of 3 possibiIiTies boTh red boTh noT and one red one noT Their probdbiIiTies odd up jusT like They should Now suppose I pick Three cors dT rondom Third verse some ds The firsT d liTTle biT longer and 0 whole loT worse Pdll Three ore red PfirsT is red X Psecond is red X PThird is red X X Pnone ore red The evenT one is red and oTher Two ore noT consisTs of R N N N R N ond N N R PR N N 03 X 07 X 07 0147 PN R N 07 x 03 x 07 PN N R 03 x 07 x 07 These ore 011 Three possible orrongemenTs of one red cor and Two noT red cors Eoch one s individuol probdbiliTy is 03 X 07 X 07 so Pone is red and Two ore noT PRNN or NRN or NNR 0147 Enough probdbiliTy of The evenT one cor is red and Two ore noT Picking cors dT rdndom from d ldrge ciTy loollting only of rednoT red ore Bernoulli Trials Bernoulli Tridls wiTh 0 fixed number n of observoTions makes The Binomiol seTTing 1 There is 0 fixed numbern of independenT Tridls For exomple suppose you pick 3 cors 2 There ore only Two ouTcomes for edch observoTion success or foilure For exomple red cor or noT red cor 3 The probdbiliTy p of success is The some for edch observoTion In 0 ldrge populdTion of cors pulling ouT one red cor does noT redlly chdnge The probdbiliTy Tth The nexT cor is red or noT i p 320 Binomidl disTribuTion Binomidl probdbiliTy model LeT X be The counT of successes ouT of 0 fixed number n of Bernoulli Triols wiTh probdbiliTy p of success on dny one Triol Then X con be 0 12 3 n X is 0 variable iT con only Take on These ceerin voluesl We soy X hos The binomiol probdbilify model wiTh pdrdmeTers n and p Suppose x is one of 0 12 n PX x con be compuTed from n p and x Soy your rdndom phenomenon is pick 0 cor dT rdndom ouT of d Idrge ciTy you ore going To pick 3 cors ond p 03 ford red cor Then The possible volues of X ore O T 2 3 ond PX x con be found from 3 03 ond x We sTorTed wiTh n 3 ond p 03 ond we worked sTep by sTep To find PX T O44T by hdnd Things were edsy when we only picked one cor Things were noT so bod when we picked Two cors Things were beginning To geT messy when we looked of Three cors thT if we picked 5 or T0 or TOO Finding The probdbiIiTy of soy 4 red cors omong TO cors picked coud geT hdiry To compuTe by hdnd T wi shorTy Turn ouT Tth we don T hove To Binomiol probobilifies NonTion for n d posiTive inTeger n fdcToriol is n n X n T X n 2 X X 3 X 2 X T Exomplez3l3X2XT6 5T5X4X3X2XTT2O TO36288OO On The Shdrp 506V see The A key in yellow MosT colculdTors wi compuTe fdcToridls wiTh d keysTroke or Two For n d posiTive inTeger k d posiTive inTeger smdller Thon n n quotCk quotn choose kquot n is colled The binomiol coefficienT kn k is The number of woys of orrdnging k successes omong n observoTions On The Shdrp 506V see The i key in yellow For exomple you ore picking 3 cors dT rdndom from d Idrge ciTy ThoT s n 3 Success is finding 0 red cor How mony woys con you find k T red cor ouT of Three observoTions We isTed 3 of Them bdck wiTh The R N N biT 3 3 3 m 3 2nd 5 T 3 woys Suppose you ore picking 5 cors How mony woys ore There To isT 2 red cors ouT of The 52 FYI The binomiol coefficienT con olso counT groups of size k from d seT of size n QThe number of 5 cord poker hdnds possible from d 52 cord deck is 52 C 5 2598960 QThe number of 4 sTudenT somples possible in d 40 sTudenT cldss is 40 C 4 p 320 box Binomiol probdbility If X hos the binomiol distribution with pdrdmeters n ond p ond x is one of X s possible volues x O or i or 2 or n then X x medns we hdd x successes so probdbility will involvep p p p x times and n x foilures so probdbility will involve l p l p l p n x times and there ore n CX woys to drrdnge x successes dnd n x foilures omong n observotions So PXx an pX l p X Q Suppose you hove n 3 cors chosen in 0 city with p 03 for red cors PX i 3 C 03 l 07 2 3XO3XO7XO7 os we hove seen O44i Nowlet s hove 0 look dt the so colled Binomiol Probobilities Toble Eoch number n of observotions hos d pordgrdph For edch n edch possible volue of x hos 0 row Possible volues of p ore in edch column Q Suppose you hove n 3 cors chosen in 0 city with p 03 for red cors You wont to know the probdbility of getting exoctly one red cor Fish out your binomiol tdble Go to pdrogroph n 3 go down to row llt i go over to the column with p 03 0t the top There we find 0441 While we re here in the binomidl setting with n 3 the possible volues for X ore X O l 2 3 ond their probdbilities ore prob Q Loollt down 0 couple of pordgrophs Suppose we pick 5 cors dt rondom The number X of red cors hos possible volues X O l 2 3 4 5 ond prob Q Suppose you ore picllting lO cors dt rdndom in 0 city with p 03 for 0 red cor tht is the probdbility thdt you find exoctly 4 red cors omong your 102 PX 4 10 C 4 034 07 6 with colculdtor or pdrogrdph n 10 row llt 4 column p 03 with binomiol probdbilities tdble p 320 again Binomial mean and standard deviation If X has the binomial distribution with parameters n and p then X is a variable one that can take on several different values some of them more likely than others The mean variance and standard deviation of X s distribution are unxp o2nXpXT p so olnxpx1 p This ONLY works for the binomial distribution Q Suppose X is binomial with n 0 and p 03 you pick 0 cars and see how many are red u 0 03 3 On average you will find 3 red cars o 1 10x03x 07 t 2 1449 with standard deviation 1449 p 321 The normal approximation to binomial distributions Sometimes even the binomial probability formula is not really practical Suppose you have a 50 question multiple choice test with 4 choices for each question You are going to guess on every single question Guessing every single time means that your answers are independent of each other The number of guesses or trialsll is fixed at n 50 On any one question there are only two outcomes either you get it right orwrong 4 choices means the probability that you guess right is p 14 025 for ea question Aha Test guessing is a binomial settingll Pyou get 30 questions right 50 C 30 025 30 075 20 0000000129 not terribly illtey What is the probability that you pass Say pass means you get 30 or more right Then Pyou pass PX Z 30 PX 30 or X 31 or X 32 or or X 50 PX 30 PX 3T PX 32 PX 50 0000000129 0000000027 0000000005 eew twenty calculations to do with calculator or software because few tables go to n 50 and a messy source of likely arithmetic error Note p 322 the successfailure condition When n X p and n X 1 p are both 2 10 the binomial distribution is approximately NormaH mean and standard deviation from n and p Does our SO guesses test fit this n X p 50 025 125 10 OK and n X 1 p 501075 375 10 also OK So the distribution of X the number of correct answers on a SO guessed questions test is approximately Normal with mean u n p 50O25 125 and standard deviation 0 ngtltpgtlt1 p J50x025x075 V9375 30618 30 125 now PX Z30 do normal calculations 2 x u W F lgtlt230Pzgt572 Let s try a different one You have been doing well in a course and you figure out that if you get 21 or more out of 50 on the Final Exam you will pass and if you get 20 or fewer you will fail What is the probability of failing the course if you guess every question Lab on Two Way Tables using Drugs data Name Open the data file drugsurvmtw and you will see the answers of 503 people 18 years or older selected at random for a phone interview in a state in the Midwest in 1997 The variables in the data file are Gender Age years of education ever smoker smoking status tried marijuana alcohol dependency Age group 1 Create a two way table for gender and having tried marijuana calculating counts and row percentages in Minitab you can use STATgtTablegtCross tabulation insert the table here a What of women have tried marijuana What of men have tried marijuana b Do you think that trying marijuana might be associated with gender c From the two way table obtain the marginal distribution of gender and write it here d From the two way table obtain the conditional distribution of marijuana conditioned to gendermale 2 Obtain the twoway table counts and row percentages for Gender amp smoking statu and paste it a Who are more likely to become smokers males or females b Who are more likely to quit smoking once they are smokers males or females 3 Obtain the twoway table for Being an ever smoker amp having tried marijuana and paste it here Are smokers more likely or less likely to try marijuana than nonsmokers 4 Obtain a two way table for Age group and alcohol dependency Which age group has the highest incidence of alcohol dependency ROBE MdTh 1530 005 009 amp 014 Fdll 2003 LecTure NoTes for The Binomial DisTribuTions pdrT of ch 17 Consider d ciTy wiTh Thousands of cors in iT Suppose Tth 30 of The cars in The ciTy ore red Now suppose you pick one cor dT rdndom from The ciTy thT is The probdbiIiTy Tth iT is 0 red cor Pred PnoT red Suppose I pick Two cors dT rdndom The cors ore independenT of one dnoTher SepdrdTer PfirsT cor is red Psecond cor is red For Two or more independenT evenTs probdbiIiTies muITipIy TogeTher PboTh cors ore red PfirsT is red and second is red PfirsT is red X Psecond is red 2 X Z PboTh ore noT red x PfirsT is red and second is noT x PfirsT is noT and second is red x Pone red in The Two corsdmple 22 Now The ided one red cor ouT of The Two chosen is on EvenT wiTh Two possible ouTcomes in iT firsT is red and second is noT or firsT is noT and second is red ProbdbiIiTies musT be colculdTed dccordingly Pone red in The Two cor sdmple P firsT is red and second is noT or firsT is noT and second is red Because these two are disjoint we can just add PfirsT is red and second is noT PfirsT is noT and second is red 2 2 Observe pick Two cors dT rdndom consisTs of 3 possibiIiTies boTh red boTh noT and one red one noT Their probdbiIiTies odd up jusT like They should Now suppose I pick Three cors dT rondom Third verse some ds The firsT d liTTle biT longer and 0 whole loT worse Pdll Three ore red PfirsT is red X Psecond is red X PThird is red X X Pnone ore red The evenT one is red and oTher Two ore noT consisTs of R N N N R N ond N N R PR N N 03 X 07 X 07 0147 PN R N 07 x 03 x 07 PN N R 03 x 07 x 07 These ore 011 Three possible orrongemenTs of one red cor and Two noT red cors Eoch one s individuol probdbiliTy is 03 X 07 X 07 so Pone is red and Two ore noT PRNN or NRN or NNR 0147 Enough probdbiliTy of The evenT one cor is red and Two ore noT Picking cors dT rdndom from d ldrge ciTy loollting only of rednoT red ore Bernoulli Trials Bernoulli Tridls wiTh 0 fixed number n of observoTions makes The Binomiol seTTing 1 There is 0 fixed numbern of independenT Tridls For exomple suppose you pick 3 cors 2 There ore only Two ouTcomes for edch observoTion success or foilure For exomple red cor or noT red cor 3 The probdbiliTy p of success is The some for edch observoTion In 0 ldrge populdTion of cors pulling ouT one red cor does noT redlly chdnge The probdbiliTy Tth The nexT cor is red or noT i p 320 Binomidl disTribuTion Binomidl probdbiliTy model LeT X be The counT of successes ouT of 0 fixed number n of Bernoulli Triols wiTh probdbiliTy p of success on dny one Triol Then X con be 0 12 3 n X is 0 variable iT con only Take on These ceerin voluesl We soy X hos The binomiol probdbilify model wiTh pdrdmeTers n and p Suppose x is one of 0 12 n PX x con be compuTed from n p and x Soy your rdndom phenomenon is pick 0 cor dT rdndom ouT of d Idrge ciTy you ore going To pick 3 cors ond p 03 ford red cor Then The possible volues of X ore O T 2 3 ond PX x con be found from 3 03 ond x We sTorTed wiTh n 3 ond p 03 ond we worked sTep by sTep To find PX T O44T by hdnd Things were edsy when we only picked one cor Things were noT so bod when we picked Two cors Things were beginning To geT messy when we looked of Three cors thT if we picked 5 or T0 or TOO Finding The probdbiIiTy of soy 4 red cors omong TO cors picked coud geT hdiry To compuTe by hdnd T wi shorTy Turn ouT Tth we don T hove To Binomiol probobilifies NonTion for n d posiTive inTeger n fdcToriol is n n X n T X n 2 X X 3 X 2 X T Exomplez3l3X2XT6 5T5X4X3X2XTT2O TO36288OO On The Shdrp 506V see The A key in yellow MosT colculdTors wi compuTe fdcToridls wiTh d keysTroke or Two For n d posiTive inTeger k d posiTive inTeger smdller Thon n n quotCk quotn choose kquot n is colled The binomiol coefficienT kn k is The number of woys of orrdnging k successes omong n observoTions On The Shdrp 506V see The i key in yellow For exomple you ore picking 3 cors dT rdndom from d Idrge ciTy ThoT s n 3 Success is finding 0 red cor How mony woys con you find k T red cor ouT of Three observoTions We isTed 3 of Them bdck wiTh The R N N biT 3 3 3 m 3 2nd 5 T 3 woys Suppose you ore picking 5 cors How mony woys ore There To isT 2 red cors ouT of The 52 FYI The binomiol coefficienT con olso counT groups of size k from d seT of size n QThe number of 5 cord poker hdnds possible from d 52 cord deck is 52 C 5 2598960 QThe number of 4 sTudenT somples possible in d 40 sTudenT cldss is 40 C 4 p 320 box Binomiol probdbility If X hos the binomiol distribution with pdrdmeters n ond p ond x is one of X s possible volues x O or i or 2 or n then X x medns we hdd x successes so probdbility will involvep p p p x times and n x foilures so probdbility will involve l p l p l p n x times and there ore n CX woys to drrdnge x successes dnd n x foilures omong n observotions So PXx an pX l p X Q Suppose you hove n 3 cors chosen in 0 city with p 03 for red cors PX i 3 C 03 l 07 2 3XO3XO7XO7 os we hove seen O44i Nowlet s hove 0 look dt the so colled Binomiol Probobilities Toble Eoch number n of observotions hos d pordgrdph For edch n edch possible volue of x hos 0 row Possible volues of p ore in edch column Q Suppose you hove n 3 cors chosen in 0 city with p 03 for red cors You wont to know the probdbility of getting exoctly one red cor Fish out your binomiol tdble Go to pdrogroph n 3 go down to row llt i go over to the column with p 03 0t the top There we find 0441 While we re here in the binomidl setting with n 3 the possible volues for X ore X O l 2 3 ond their probdbilities ore prob Q Loollt down 0 couple of pordgrophs Suppose we pick 5 cors dt rondom The number X of red cors hos possible volues X O l 2 3 4 5 ond prob Q Suppose you ore picllting lO cors dt rdndom in 0 city with p 03 for 0 red cor tht is the probdbility thdt you find exoctly 4 red cors omong your 102 PX 4 10 C 4 034 07 6 with colculdtor or pdrogrdph n 10 row llt 4 column p 03 with binomiol probdbilities tdble p 320 again Binomial mean and standard deviation If X has the binomial distribution with parameters n and p then X is a variable one that can take on several different values some of them more likely than others The mean variance and standard deviation of X s distribution are unxp o2nXpXT p so olnxpx1 p This ONLY works for the binomial distribution Q Suppose X is binomial with n 0 and p 03 you pick 0 cars and see how many are red u 0 03 3 On average you will find 3 red cars o 1 10x03x 07 t 2 1449 with standard deviation 1449 p 321 The normal approximation to binomial distributions Sometimes even the binomial probability formula is not really practical Suppose you have a 50 question multiple choice test with 4 choices for each question You are going to guess on every single question Guessing every single time means that your answers are independent of each other The number of guesses or trialsll is fixed at n 50 On any one question there are only two outcomes either you get it right orwrong 4 choices means the probability that you guess right is p 14 025 for ea question Aha Test guessing is a binomial settingll Pyou get 30 questions right 50 C 30 025 30 075 20 0000000129 not terribly illtey What is the probability that you pass Say pass means you get 30 or more right Then Pyou pass PX Z 30 PX 30 or X 31 or X 32 or or X 50 PX 30 PX 3T PX 32 PX 50 0000000129 0000000027 0000000005 eew twenty calculations to do with calculator or software because few tables go to n 50 and a messy source of likely arithmetic error Note p 322 the successfailure condition When n X p and n X 1 p are both 2 10 the binomial distribution is approximately NormaH mean and standard deviation from n and p Does our SO guesses test fit this n X p 50 025 125 10 OK and n X 1 p 501075 375 10 also OK So the distribution of X the number of correct answers on a SO guessed questions test is approximately Normal with mean u n p 50O25 125 and standard deviation 0 ngtltpgtlt1 p J50x025x075 V9375 30618 30 125 now PX Z30 do normal calculations 2 x u W F lgtlt230Pzgt572 Let s try a different one You have been doing well in a course and you figure out that if you get 21 or more out of 50 on the Final Exam you will pass and if you get 20 or fewer you will fail What is the probability of failing the course if you guess every question

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.