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The problems considered above should be interesting and practical for our students I think our students should have mastery of these problems and others like them since data of this kind are frequently encountered in various elds Mathematically, the analyses given above are equivalent to those found by using a probability theorem known as Bayes theorem The geometric model given here shows that this result need not be known since it follows so simply from the area of a rectangle The analysis given here should make these problems accessible to elementary students of mathematics
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1 Three methods, say A, B, and C, are sometimes used to teach an industrial worker a skill The methods fail to instruct with rates of 20%, 10%, and 30%, respectively Cost considerations restrict method A to be used twice as often as B that is used twice as often as C If a worker fails to learn the skill, what is the probability that she was taught by method A 2 Binary symbols (0 or 1) sent over a communication line are sometimes interchanged The probability that a 0 is changed to 1 is 01 while the probability that a 1 is changed to 0 is 02 The probability that a 0 is sent is 04 and the probability that a 1 is sent is 06 If a 1 is received, what is the probability that a 0 was sent 3 Sample surveys are often subject to error because the respondent might not truthfully answer a sensitive question such as Do you use illegal drugs A procedure known as randomized response is sometimes used Here is how that works A respondent is asked to ip a coin and not reveal the result If the coin comes up heads, the respondent answers the sensitive question, otherwise he
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responds to an innocuous question such as Is your Social Security number even So if the respondent answers Yes, it is not known to which question he is responding Show, however, that with a large number of respondents, the frequency of illegal drug use can be determined 4 Combine cards from several decks and create another deck of cards with, say, 12 aces and 40 other cards Have students select a card and not reveal whether it is an ace or not If an ace is chosen, have the students answer the question about illegal drugs in the previous exploration and otherwise answer an innocuous question Then approximate the use of illegal drugs 5 A certain brand of lie detector is accurate with probability 092; that is, if a person is telling the truth, the detector indicates he is telling the truth with probability 092, while if he is lying, the detector indicates he is lying with probability 092 Assume that 98% of the subjects of the test are truthful What is the probability that a person is lying if the detector indicates he is lying
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Geometric Probability
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to use connections between geometry and probability to see how the solution of a quadratic equation solves a geometric problem to use linear inequalities in geometric problems to show some unusual problems for geometry
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EXAMPLE 41
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Meeting at the Library
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Joan and Jim agree to meet at the library after school between 3 and 4 pm Each agrees to wait no longer than 15 min for the other What is the probability that they will meet This at rst glance does not appear to be a geometric problem, but it is We show the situation in Figure 41 For convenience, we take the interval between 3 and 4 pm to be the interval between 0 and 1, so both Joan and Jim s waiting time becomes 1/4 of an hour We suppose that each person arrives at some random time, so these arrival times are points somewhere in a square of side 1 Let X denote Joan s arrival time and Y denote Jim s arrival time If Joan arrives rst, then Jim s arrival time Y must be greater than Joan s arrival time X So Y > X or Y X > 0 and if they are to meet, then Y X < 1/4 or Y < X + 1/4 This is the region below the line Y = X + 1/4 and has intercepts at (0, 1/4) and (3/4, 1) and is the top line in Figure 41 If Jim arrives rst, then X > Y and X Y > 0 and if they are to meet, then X Y < 1/4 or Y > X 1/4 This is the region above the line Y = X 1/4 and has intercepts at (1/4, 0) and (1, 3/4) and is the lower line in Figure 41 They both meet then when Y < X + 1/4 and when Y > X 1/4 This is the region between the lines and is shaded in the gure Since the area of the square is 1, the shaded area must represent the probability that Joan and Jim meet The easiest way to compute this is to subtract the areas of the two triangles
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