Sectin 8.1: The Binmial Distributins Chapter 8: The Binmial and Gemetric Distributins A randm variable X is called a BINOMIAL RANDOM VARIABLE if it meets ALL the fllwing cnditins: 1) 2) 3) 4) The MOST IMPORTANT skill fr using Binmial Distributins is the ability t where it des and des nt apply. Given that X is a cunt. Des X have a binmial distributin? Ex 1) The pl f ptential jurrs fr a murder case cntains 100 persns chsen at randm frm the adult residents f a large city. Each persn in the pl is asked whether he r she ppses the death penalty: X is the number wh say yes. 1) Fixed number f trials? 2) Each trial independent? 3) Tw disjint utcmes (success & failure)? 4) Each trial, prbability f success is the same? Ex 2) At peak perids, 15% f attempted lg-ins t an email service fail. Lg-in attempts are independent and each has the same prbability f failing. Christian lgs in repeatedly until he succeeds. X is the number f the lg-in attempt that finally succeeds. 1) Each trial independent? 2) Tw disjint utcmes (success & failure)? 3) Each trial, prbability f success is the same? 4) Fixed number f trials? Definitin f Binmial Distributin: The distributin f the cunt X f in the binmial setting is the binmial distributin with parameters (number f ) and (the f success n any ). The pssible values f X are the frm t. We say that X is B(n, p)
Chapter 8: The Binmial and Gemetric Distributins The randm variable X is the cunt f the number f successes in the n trials. The prbability that exactly k successes are btained in n trials f a binmial randm variable is, where is called the binmial cefficient and has a value Example 1: The prbability f a thumbtack landing pints up when tssed is 0.42. If a thumbtack is tssed 8 times, what is the prbability that it lands pints up exactly twice? 1 st ) Check t see if this is a binmial setting: a) Fixed number f trials? Yes 8 b) Independent the utcme f ne tack tss has nthing t d with the utcme f the next tack tss. c) Exactly tw utcmes, success = pint up, failure = pint dwn. d) Prb. f success is cnstant =.42. yes, we have a binmial setting prceed = = = 28 (0.42) 2 (.58) 6 =.1888 Hwever, binmial prbabilities can be calculated using the graphing calculatr. Fr P(X = k) using the calculatr: 2 nd VARS binmpdf (n,p, k) where n = number f p = f k = number f binmpdf (8,.42, 2) = Example 2: The prbability f a thumbtack landing pints up when tssed is 0.42. If a thumbtack is tssed 8 times, what is the prbability that it lands pints up at mst three times? 1 st ) This is a binmial setting, because the 4 cnditins listed abve are satisfied. Answer: The prbability f at mst tw f 8 pints up landings is P(x < 3) = + + + Cummulative binmial prbabilities can be calculated using the graphing calculatr t! Fr P(k X) using the calculatr: 2 nd VARS binmcdf(n, p, k) where n = nbr. f trials, p = prb. f success, and x = nbr. f successes r less binmcdf (8,.42, 3) =
Chapter 8: The Binmial and Gemetric Distributins Smetimes it is kay t assume a binmial setting even thugh the independence cnditin is satisfied. (Study Ex. 8.3 Inspecting Switches, p. 515) An engineer chses a simple randm sample f 10 switches frm a shipment f 10,000 switches. Assuming that a defective switch is drawn first (p = 1/10,000), the prbability fr the 2 nd switch f being defective changes t (p = 1/ ). Hwever, remving 1 switch frm a shipment f 10,000 changes the makeup f the remaining 9999 switches very little. S even thugh the cnditin f des nt strictly hld, fr practical purpses, this behaves like a binmial setting. Sampling Distributin f Cunt: Chse an SRS f size n frm a ppulatin with prprtin p f successes. When the ppulatin is than the sample, the cunt X f successes in the sample has apprximately the binmial distributin with parameters n and p. If X is B(n, p), find µ x and x (that is, calculate the mean and variance f a binmial distributin). Mean and Standard Deviatin: (number f trials)(prbability f success) Vcabulary: Binmial Setting randm variable meets binmial cnditins Trial each repetitin f an experiment Success ne assigned result f a binmial experiment Failure the ther result f a binmial experiment PDF prbability distributin functin; assigns a prbability t each value f X CDF cumulative (prbability) distributin functin; assigns the sum f prbabilities less than r equal t X Binmial Cefficient cmbinatin f k success in n trials Factrial n! is n (n-1) (n-2) 2 1
Chapter 8: The Binmial and Gemetric Distributins Example 1: Des this setting fit a binmial distributin? Explain a) NFL kicker has made 80% f his field gal attempts in the past. This seasn he attempts 20 field gals. The attempts differ widely in distance, angle, wind and s n. 1) Fixed nbr. trials? 2) Independence? 3) Prb. f success & failure? 4) Prb. f success is equal fr all trials? b) NBA player has made 80% f his ful shts in the past. This seasn he takes 150 free thrws. Basketball free thrws are always attempted frm 15 ft away with n interference frm ther players. 1) Fixed nbr. trials? 2) Independence? 3) Prb. f success & failure? 4) Prb. f success is equal fr all trials? Example 2: In the Pepsi Challenge a randm sample f 20 subjects are asked t try tw unmarked cups f pp (Pepsi and Cke) and chse which ne they prefer. If preference is based slely n chance what is the prbability that: a) 6 will prefer Pepsi? b) 12 will prefer Cke? c) at least 15 will prefer Pepsi? d) at mst 8 will prefer Cke? Example 3: A certain medical test is knwn t detect 90% f the peple wh are afflicted with disease Y. If 15 peple with the disease are administered the test what is the prbability that the test will shw that: a) all 15 have the disease? b) at least 13 peple have the disease? c) 8 have the disease?
Chapter 8: The Binmial and Gemetric Distributins Example 4: Find the mean and standard deviatin f a binmial distributin with n = 10 and p = 0.1 Example 5: Sample surveys shw that fewer peple enjy shpping than in the past. A survey asked a natinwide randm sample f 2500 adults if shpping was ften frustrating and time-cnsuming. Assume that 60% f all US adults wuld agree if asked the same questin, what is the prbability that 1520 r mre f the sample wuld agree? Example 6: Each entry in a table f randm digits like Table B in ur bk has a prbability f 0.1 f being a zer. a) Find prbability f find exactly 4 zers in a line 40 digits lng. b) What is the prbability that a grup f five digits frm the table will cntain at least 1 zer? Example 7: A university claims that 80% f its basketball players get their degree. An investigatin examines the fates f a randm sample f 24 players wh entered the prgram ver a perid f several years. Of these players, 12 graduated and 12 are n lnger in schl. If the university's claim is true, a) What is the prbability that exactly 12 ut f 24 graduate? b) What is the prbability that all 24 graduate?