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~~7-1 . ~~
~~Examples of binomial experiments ~~
~~Some experiments are composed of repetitions of independent trials , each with two possible outcomes . ~~
~~The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another . ~~
~~We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena , but also because it has important properties that recur in many other probability models . ~~
~~We begin with a few examples of binomial experiments . ~~
~~Marksmanship example . ~~

~~A trained marksman shooting five rounds at a target , all under practically the same conditions , may hit the bull's-eye from 0 to 5 times . ~~
~~In repeated sets of five shots his numbers of bull's-eyes vary . ~~
~~What can we say of the probabilities of the different possible numbers of bull's-eyes ? ? ~~
~~Inheritance in mice . ~~

~~In litters of eight mice from similar parents , the number of mice with straight instead of wavy hair is an integer from 0 to 8 . ~~
~~What probabilities should be attached to these possible outcomes ? ? ~~
~~Aces ( ones ) with three dice . ~~

~~When three dice are tossed repeatedly , what is the probability that the number of aces is 0 ( or 1 , or 2 , or 3 ) ? ? ~~
~~General binomial problem . ~~

~~More generally , suppose that an experiment consists of a number of independent trials , that each trial results in either a `` success '' or a `` non-success '' ( `` failure '' ) , and that the probability of success remains constant from trial to trial . ~~
~~In the examples above , the occurrence of a bull's-eye , a straight-haired mouse , or an ace could be called a `` success '' . ~~
~~In general , any outcome we choose may be labeled `` success '' . ~~

~~The major question in this chapter is : What is the probability of exactly X successes in N trials ? ? ~~

~~In Chapters 3 and 4 we answered questions like those in the examples , usually by counting points in a sample space . ~~
~~Fortunately , a general formula of wide applicability solves all problems of this kind . ~~
~~Before deriving this formula , we explain what we mean by `` problems of this kind '' . ~~

~~Experiments are often composed of several identical trials , and sometimes experiments themselves are repeated . ~~
~~In the marksmanship example , a trial consists of `` one round shot at a target '' with outcome either one bull's-eye ( success ) or none ( failure ) . ~~
~~Further , an experiment might consist of five rounds , and several sets of five rounds might be regarded as a super-experiment composed of several repetitions of the five-round experiment . ~~
~~If three dice are tossed , a trial is one toss of one die and the experiment is composed of three trials . ~~
~~Or , what amounts to the same thing , if one die is tossed three times , each toss is a trial , and the three tosses form the experiment . ~~
~~Mathematically , we shall not distinguish the experiment of three dice tossed once from that of one die tossed three times . ~~
~~These examples are illustrative of the use of the words `` trial '' and `` experiment '' as they are used in this chapter , but they are quite flexible words and it is well not to restrict them too narrowly . ~~
~~Example 1 . ~~
~~Student football managers . ~~

~~Ten students act as managers for a high-school football team , and of these managers a proportion P are licensed drivers . ~~
~~Each Friday one manager is chosen by lot to stay late and load the equipment on a truck . ~~
~~On three Fridays the coach has needed a driver . ~~
~~Considering only these Fridays , what is the probability that the coach had drivers all 3 times ? ? ~~
~~Exactly 2 times ? ? ~~
~~1 time ? ? ~~
~~0 times ? ? ~~
~~Discussion . ~~

~~Note that there are 3 trials of interest . ~~
~~Each trial consists of choosing a student manager at random . ~~
~~The 2 possible outcomes on each trial are `` driver '' or `` nondriver '' . ~~
~~Since the choice is by lot each week , the outcomes of different trials are independent . ~~
~~The managers stay the same , so that Af is the same for all weeks . ~~
~~We now generalize these ideas for general binomial experiments . ~~

~~For an experiment to qualify as a binomial experiment , it must have four properties : ( 1 ) there must be a fixed number of trials , ( 2 ) each trial must result in a `` success '' or a `` failure '' ( a binomial trial ) , ( 3 ) all trials must have identical probabilities of success , ( 4 ) the trials must be independent of each other . ~~
~~Below we use our earlier examples to describe and illustrate these four properties . ~~
~~We also give , for each property , an example where the property is absent . ~~
~~The language and notation introduced are standard throughout the chapter . ~~
~~1 . ~~
~~There must be a fixed number n of repeated trials . ~~

~~For the marksman , we study sets of five shots ( Af ) ; ; ~~
~~for the mice , we restrict attention to litters of eight ( Af ) ; ; ~~
~~and for the aces , we toss three dice ( Af ) . ~~
~~Experiment without a fixed number of trials . ~~

~~Toss a die until an ace appears . ~~
~~Here the number of trials is a random variable , not a fixed number . ~~
~~2 . ~~
~~Binomial trials . ~~

~~Each of the N trials is either a success or a failure . ~~
~~`` Success '' and `` failure '' are just convenient labels for the two categories of outcomes when we talk about binomial trials in general . ~~
~~These words are more expressive than labels like `` A '' and `` not-A '' . ~~
~~It is natural from the marksman's viewpoint to call a bull's-eye a success , but in the mice example it is arbitrary which category corresponds to straight hair in a mouse . ~~
~~The word `` binomial '' means `` of two names '' or `` of two terms '' , and both usages apply in our work : the first to the names of the two outcomes of a binomial trial , and the second to the terms P and Af that represent the probabilities of `` success '' and `` failure '' . ~~
~~Sometimes when there are many outcomes for a single trial , we group these outcomes into two classes , as in the example of the die , where we have arbitrarily constructed the classes `` ace '' and `` not-ace '' . ~~
~~Experiment without the two-class property . ~~

~~We classify mice as `` straight-haired '' or `` wavy-haired '' , but a hairless mouse appears . ~~
~~We can escape from such a difficulty by ruling out the animal as not constituting a trial , but such a solution is not always satisfactory . ~~
~~3 . ~~
~~All trials have identical probabilities of success . ~~

~~Each die has probability Af of producing an ace ; ; ~~
~~the marksman has some probability p , perhaps 0.1 , of making a bull's-eye . ~~
~~Note that we need not know the value of p , for the experiment to be binomial . ~~
~~Experiment where p is not constant . ~~

~~During a round of target practice the sun comes from behind a cloud and dazzles the marksman , lowering his chance of a bull's-eye . ~~
~~4 . ~~
~~The trials are independent . ~~

~~Strictly speaking , this means that the probability for each possible outcome of the experiment can be computed by multiplying together the probabilities of the possible outcomes of the single binomial trials . ~~
~~Thus in the three-dice example Af , Af , and the independence assumption imply that the probability that the three dice fall ace , not-ace , ace in that order is Af . ~~
~~Experimentally , we expect independence when the trials have nothing to do with one another . ~~
~~Examples where independence fails . ~~

~~A family of five plans to go together either to the beach or to the mountains , and a coin is tossed to decide . ~~
~~We want to know the number of people going to the mountains . ~~
~~When this experiment is viewed as composed of five binomial trials , one for each member of the family , the outcomes of the trials are obviously not independent . ~~
~~Indeed , the experiment is better viewed as consisting of one binomial trial for the entire family . ~~
~~The following is a less extreme example of dependence . ~~
~~Consider couples visiting an art museum . ~~
~~Each person votes for one of a pair of pictures to receive a popular prize . ~~
~~Voting for one picture may be called `` success '' , for the other `` failure '' . ~~
~~An experiment consists of the voting of one couple , or two trials . ~~
~~In repetitions of the experiment from couple to couple , the votes of the two persons in a couple probably agree more often than independence would imply , because couples who visit the museum together are more likely to have similar tastes than are a random pair of people drawn from the entire population of visitors . ~~
~~Table 7-1 illustrates the point . ~~
~~The table shows that 0.6 of the boys and 0.6 of the girls vote for picture A . ~~
~~Therefore , under independent voting , Af or 0.36 of the couples would cast two votes for picture A , and Af or 0.16 would cast two votes for picture B . ~~
~~Thus in independent voting , Af or 0.52 of the couples would agree . ~~
~~But Table 7-1 shows that Af or 0.70 agree , too many for independent voting . ~~

~~Each performance of an n-trial binomial experiment results in some whole number from 0 through N as the value of the random variable X , where Af . ~~
~~We want to study the probability function of this random variable . ~~
~~For example , we are interested in the number of bull's-eyes , not which shots were bull's-eyes . ~~
~~A binomial experiment can produce random variables other than the number of successes . ~~
~~For example , the marksman gets 5 shots , but we take his score to be the number of shots before his first bull's-eye , that is , 0 , 1 , 2 , 3 , 4 ( or 5 , if he gets no bull's-eye ) . ~~
~~Thus we do not score the number of bull's-eyes , and the random variable is not the number of successes . ~~
~~The constancy of P and the independence are the conditions most likely to give trouble in practice . ~~
~~Obviously , very slight changes in P do not change the probabilities much , and a slight lack of independence may not make an appreciable difference . ~~
~~( For instance , see Example 2 of Section 5-5 , on red cards in hands of 5 . ~~
~~) On the other hand , even when the binomial model does not describe well the physical phenomenon being studied , the binomial model may still be used as a baseline for comparative purposes ; ; ~~
~~that is , we may discuss the phenomenon in terms of its departures from the binomial model . ~~
~~To summarize : ~~
~~A binomial experiment consists of Af independent binomial trials , all with the same probability Af of yielding a success . ~~
~~The outcome of the experiment is X successes . ~~
~~The random variable X takes the values Af with probabilities Af or , more briefly Af . ~~

~~We shall find a formula for the probability of exactly X successes for given values of P and N . ~~
~~When each number of successes X is paired with its probability of occurrence Af , the set of pairs Af , is a probability function called a binomial distribution . ~~
~~The choice of P and N determines the binomial distribution uniquely , and different choices always produce different distributions ( except when Af ; ; ~~
~~then the number of successes is always 0 ) . ~~
~~The set of all binomial distributions is called the family of binomial distributions , but in general discussions this expression is often shortened to `` the binomial distribution '' , or even `` the binomial '' when the context is clear . ~~
~~Binomial distributions were treated by James Bernoulli about 1700 , and for this reason binomial trials are sometimes called Bernoulli trials . ~~
~~Random variables . ~~

~~Each binomial trial of a binomial experiment produces either 0 or 1 success . ~~
~~Therefore each binomial trial can be thought of as producing a value of a random variable associated with that trial and taking the values 0 and 1 , with probabilities Q and P respectively . ~~
~~The several trials of a binomial experiment produce a new random variable X , the total number of successes , which is just the sum of the random variables associated with the single trials . ~~
~~Example 2 . ~~

~~The marksman gets two bull's-eyes , one on his third shot and one on his fifth . ~~
~~The numbers of successes on the five individual shots are , then , 0 , 0 , 1 , 0 , 1 . ~~
~~The number of successes on each shot is a value of a random variable that has values 0 or 1 , and there are 5 such random variables here . ~~
~~Their sum is X , the total number of successes , which in this experiment has the value Af . ~~