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EXAMPLE 13
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A box contains slips of paper numbered 1, 2, and 3, respectively Slips are drawn one at a time, replaced, and a cumulative running sum is kept until the sum equals or exceeds 4 This is an example of a waiting time problem; we wait until an event occurs The event can occur in two, three, or four drawings (It must occur no later than the fourth drawing) The sample space is shown in Table 14, where n is the number of drawings and the sample points show the order in which the integers were selected Table 14 n 2 3 4 Orders (1,3),(3,1),(2,2) (2,3),(3,2),(3,3) (1,1,2),(1,1,3),(1,2,1),(1,2,2) (1,2,3),(2,1,1),(2,1,2),(2,1,3) (1,1,1,1),(1,1,1,2),(1,1,1,3)
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Probability and Sample Spaces Table 15 n 1 2 3 4 5 Expected value 200 225 237 244 249
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We will show later that the expected number of drawings is 237 What happens as the number of slips of paper increases The approach used here becomes increasingly dif cult Table 15 shows exact results for small values of n, where we draw until the sum equals or exeeds n + 1 While the value of n increases, the expected length of the game increases, but at a decreasing rate It is too dif cult to show here, but the expected length of the game approaches e = 271828 as n increases This does, however, make a very interesting classroom exercise either by generating random numbers within the speci ed range or by a computer simulation The result will probably surprise students of calculus and be an interesting introduction to e for other students
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EXAMPLE 14
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An In nite Sample Space
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Examples 11, 12, and 13 are examples of nite sample spaces, since they contain a nite number of elements We now consider an in nite sample space We observe a production line until a defective (D) item appears The sample space now is in nite since the event may never occur The sample space is shown below (where G denotes a good item)
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We note that S in this case is a countable set, that is, a set that can be put in one-to-one correspondence with the set of positive integers Countable sample spaces often behave as if they were nite Uncountable in nite sample spaces are also encountered in probability, but we will not consider these here
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Sample Spaces
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EXAMPLE 15
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Tossing a Coin
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We toss a coin ve times and record the tosses in order Since there are two possibilities on each toss, there are 25 = 32 sample points A sample space is shown below
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TTTTT HHTTT
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TTTTH HTHTT HTTTH
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TTHTT TTTHH TTHTH HTHTH HHHTH
THTTT THTHT HHHHT THHTT HHTHH
HTTHH
THTHH
HHHTT HTHHH
HTTTT TTHHT
THHTH THHHT TTHHH HTHHT HHTHT HHHHT THHHH HHHHH
It is also possible in this example simply to count the number of heads, say, that occur In that case, the sample space is S1 = {0, 1, 2, 3, 4, 5} Both S and S1 are sets that contain all the possibilities when the experiment is performed and so are sample spaces So we see that the sample space is not uniquely de ned Perhaps one can think of other sets that describe the sample space in this case
EXAMPLE 16
AP Statistics
A class in advanced placement statistics consists of three juniors (J) and four seniors (S) It is desired to select a committee of size two An appropriate sample space is S = {JJ, JS, SJ, SS} where we have shown the class of the students selected in order One might also simply count the number of juniors on the committee and use the sample space S1 = {0, 1, 2} Alternatively, one might consider the individual students selected so that the sample space, shown below, becomes S2 = {J1 J2 , J1 J3 , J2 J3 , S1 S2 , S1 S3 , S1 S4 , S2 S3 , S2 S4 , S3 S4 , J1 S1 , J1 S2 , J1 S3 , J1 S4 , J2 S1 , J2 S2 , J2 S3 , J2 S4 , J3 S1 , J3 S2 , J3 S3 , J3 S4 } S2 is as detailed a sample space one can think of, if order of selection is disregarded, so one might think that these 21 sample points are equally likely to occur provided no priority is given to any of the particular individuals So we would expect that each of the points in S2 would occur about 1/21 of the time We will return to assigning probabilities to the sample points in S and S2 later in this chapter