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AP Stats / CH 8: Binomial vs Geometric Distributions Distribution Every variable has a distribution, whether the variable is categorical or numerical, continuous

s or discreet. A variables distribution consists of the values it takes on and how often it takes each of these values. Distributions are often shown as graphs so you can see the relative frequency with which values occur. Tables are used, too. Properties of a binomial distribution: #1 ! are t"e ! conditions t"at indicate a binomial distribution #a class of discrete probabilit\$ distributions% ! #! \$! %! There are n repeated trials " a fi&ed number of observations All trials are identical and independent The probability of success is the same for each observation Each trial has e&actl\$ t'o possible outcomes, &success' and &failure'

()success in a given trial! * p ()failure in a given trials! * " p +f , * the number of successes in n trials * -, , #, ., n , is /)n, p! 0000 n observations, probability p, k successes (),*k! *

n k nk k p (1 p)

...

= E ( x) = np 2 3ariance of the binomial random variable2 = np (1 p )

1ean of the binomial random variable2 T+4\$2 binompdf)n, p, ,! 00 under Distr. )pdf * probability distribution function! T+452 tistat.binom(df)n, p, ,! 000in 6ATA789 under :lash Apps (), n! * sum of each probability from n and below (),;n! * " (), n!

almost binomial (a.k.a. binomial settin(% As a rule of thumb, we consider a situation to be almost binomial when the population is at least #- times the sample si<e. )=ome te>tbooks say - times the sample si<e.! +n terms of statistical analysis, almost binomial situations can be treated as though they are binomial. Binomial )ormal

&As the number of trials n gets larger, the binomial distribution gets closer to the normal distribution. ?hen n is large and the binomial calculations are cumbersome, we can use normal probability calculations to appro>imate the binomial probabilities. Rule of thumb: @ou can use normal appro>. if 2

np 10 and n(1 p ) 10

Properties of a (eometric distribution: the trials are repeated until the first success is observed All trials are identical and independent Each trial has two possible outcomes, &success' and &failure' ()success in a given trial! * p ()failure in a given trials! * " p

1 p (1 p) 2 = p2 n (),;n! * (1 p )

)prob. that it takes more than n trials for a success!

Binomial or (eometric or neit"er* . A worker opening oysters to look for pearls counts the number of oysters he has to open until he finds the first pearl. #. A supervisor at the end of an assembly line counts the number if non0defective items produced until he finds a defective one. \$. Deal - cards. 6ount the number , of red cards. %. A quality control inspector takes a random sample of #- items from a large lot, inspects each item, classifies each as defective or non0defective, and counts the number of defective items in the sample. A. An engineer chooses and =B= of - switches from a shipment of -,--- switches. =uppose -C of the switches in the shipment are bad. The engineer counts the number , of bad switches in the sample. D. A homeowner buys \$- a<alea plants from a nursery. The number if plants that survive at the end of the year are counted. E. An electrician inspecting cable one yard at a time for defects counts the number if yards he inspects before he finds a defect. 4. A game requires you to roll a pair of dice until you get doubles. 5. A teacher asks you to roll a pair of dice - times and record the number of times you got doubles.

/+F81+A7 +&, 1 6orinne is a basketball player who makes EAC of her free throws over the course of a season. +n a key game, 6orinne shoots # free throws and makes only E of them. +s this unusual for 6orinneG The H of , baskets in # attempts has the /) #, -.EA! distribution. :ind (),*E!

:ind (),

7)

+-, . Each child born to a particular set of parents has probability -.#A of having /lood type 8. =uppose these parents have A children and we are interested in the number of children who have type 8 blood. Thus, , is /)A, -.#A!. a! ?hat is the probability that e>actly # of the children have type 8 bloodG

+-, / An =B= of - switches from a shipment of -,--- switches as probability . of being bad. a! :ind the probability that no more than switch is bad.

b! :ind the mean and standard deviation.

+&, !, +n a recent survey #A-- people were asked if shopping has become frustrated. =uppose that in actuality D-C of I= adults &Agree'. a! ?hat is the probability that A#- or more of the sample agreeG

+-, 0 =uppose a family eats at a fast food restaurant. There are three possible toys )a car, a top, or a yo0yo! given with the kids meals. The toys are placed in the meal bags at random. =uppose this family buys a kids meal at this restaurant on four different days. a. ?hat is the probability that \$ out of % meals will come with a yo0yoG b. ?hat is the probability that at most # meals will come with a yo0yoG c. ?hat is the probability of getting at least \$ yo0yos with % meals purchasedG d. ?hat is the e>pected number of yo0yos when % meals are purchasedG e. Jow can we simulate this processG

f. 7ets repeat it -- times and record the results. 1ean of the geometric random variable2

= E ( x) =

1 p p

3ariance of the geometric random variable2

2 =

(1 p) 2 p

g. Jow can we turn this kids meal problem into a geometric distributionG

h. :ind the probability that the family will have to buy \$ meals to get a yo0yo. i. :ind the probability that the family will have to buy A meals to get a yo0yo. K. ?hat is the e>pected number of meals needed to get a yo0yoG

l. Bepeat the process -- times. Becord the results.

A probabilit\$ e&periment is an e&periment '"ose possible outcomes ma\$ be 1no'n but '"ose e&act outcome is a random event and cannot be predicted 'it" certaint\$, Binomial Distributions +n order to use a sample value to make an inference about a population value, you use a probability distribution. A probability distribution tells you whether your sample value is common, or whether itLs a value youLd almost never e>pect to get. +f you find that your sample value is e>tremely rare according to your probability distribution, you may have reason to doubt some assumptions you made about the population. This lesson looks at the binomial distribution, a probability distribution useful for making many types of inferences. This distribution is useful when modeling the likelihood of events either succeeding of failing independently, such as voters voting for or against a ballot measure. Inder certain conditions, you can use the normal distribution as a model for the binomial distribution. This comes in handy, since you can use z0scores to find probabilities like youLd do with a normal distribution.

2b3ectives /y the end of this lesson you should be able to2 7ist the four conditions that define a binomial setting, and use those conditions to determine whether a situation is binomial. =olve binomial problems involving at least, at most, and between. Ise the normal appro>imation in binomial setting problems. A(E, Fotes binomcdf / binomial probabilities for a ran(e of outcomes /inomial cumulative density function. The function on the T+04\$ calculator that finds the probability for a range of successes within n trials.

The synta> is binomcdf)n,p,>!, where n is the number of trials, p is the probability of success for each trial, and > is the upper bound for the range of successes. :or e>ample, binomcdf#.45,6058% will find the probability of getting 4 or fewer successes in a situation with #- trials and a probability of success for each trial of . EA. Access this function by pressing #nd MD+=TBN, and scrolling down to binomcdf). Fote that since the binomial distribution is discrete, ()> O %! would have an upper bound of \$ )you are not including the %!, but ()> %! would have an upper bound of % )since you would include the %!. Therefore, if you want to calculate (), O %!, the synta> is binomcdf)#-,.EA,\$!. binomial e&periment ?hen we count the number of successful outcomes in a binomial event, we refer to this as a binomial e>periment. binomial probabilities for a ran(e of outcomes The probability of having a range of outcomes rather than a single value. :or e>ample, the probability of having less than or equal to > successes out of n trials from a binomial distribution /)n, p!. The formula for calculating such a probability is (), >! * (),* ! P (),*#! P ... P (),*>!. +n order to do these computations, we most often use the binomcdf)n,p,>! function found in the D+=TB menu of the T+04\$. binompdf / binomial probabilities for an e&act number of outcomes

(robability of having e>actly > successes out of n trials from a binomial distribution /)n, p! The formula for calculating such probability is (),*>!* p > ) 0p! n0>,

,*-, , ... , n,where gives the number of ways to select > things from n things. The T+04\$ computes this as binompdf)n,p,>!, which can be found in the D+=TB menu of your calculator. Binomial probabilit\$ densit\$ function. The function on the T+04\$ calculator that finds the probability for an e>act number of successes within n trials. The synta> is binompdf )n,p,>!, where n is the number of trials, p is the probability of success for each trial, and > is the number of successes. Access this function by pressing #nd MD+=TBN, and scrolling down to binompdf). Continuit\$ correction ?hen using the normal appro>imation to the binomial, you can increase accuracy by treating values as though they include the .A below and above them. +n other words, include .A below the lower bound, and .A above the upper bound. /e sure to use the true upper or lower bound. :or instance, if you are looking for ()> O %!, your upper bound is \$ because your interval does not include %. =o to use the continuity correction you would find ()> #.A! +f you are looking for ()> %!, then your upper

bound does include %, and to use the continuity correction you would find ()> \$.A!, Fote that if your upper or lower bound is infinity, you only need to add .A to the bound that is not infinity.

binomial settin( A binomial setting has the following properties2 . There is a set number, n, of identical trials #. The outcome of each trial is either a success or a failure. \$. The probability of success on a single trial is p. %. The trials are independent. +n a binomial e>periment, you are interested in the probability of getting > successes in n trials. (eometric settin( A geometric setting has the following properties2 The trials are independent. The outcome of each trial is either success or failure. (robability of success on a single trial is p. normal appro&imation to t"e binomial +f , follows the /)n, p! distribution, and if np - and n) 0 p! - many te>tbooks use A instead of -!, then , follows an appro>imately normal distribution, with mean and standard deviation2 Binomial Situations #+vents% /y now in your study of statistics, you should be able to answer such questions as, ?hatLs the probability of getting e>actly three heads on five flips of a coinG or ?hatLs the probability of getting a value between 4 and - from a normal distribution with a mean of and a standard deviation of #G

average waiting time Average number of trials until you get the first success. :or a geometric setting, average waiting time is always Qp. (eometcdf / Geometric cumulative densit\$ function, The function on the T+04\$ calculator that finds the probability in a geometric setting of getting the first success in > trials or less. The synta> is geometcdf)p,>!, where p is the probability of success for each trial, and > is the upper bound for the range of the number of trials. :or e>ample, geometcdf).EA,4! will find the probability of getting your first success in 4 trials or less, if the probability of success on each trial is .EA. Access this function by pressing #nd MD+=TBN, and scrolling down to geometcdf).

Fote that since the geometric distribution is discrete, ()> O %! has an upper bound of \$ )you do not include the %!, but ()> %! has an upper bound of % )since you include the %!. Thus if you want to calculate (),, %!, the synta> is geometcdf).EA,\$!. (eometric distribution / (eometric settin( A distribution of the number of trials required to obtain the first success under a geometric setting. =uppose , is a random variable with a geometric distribution and has probability of success p on each trial. Then , takes the values , #, \$, ..., and (), * n! * ) 0 p!n0pp.
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