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John Paulos: INNUMERACY: Mathematical Illiteracy and Its Consequences

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John Paulos INNUMERACY: Mathematical Illiteracy and Its Consequences

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Paulos speaks mainly of the dangers of mathematical innumeracy; that is, the common misconceptions of the layperson in regards to numbers, exploring the relationship between math and the human mind. Paulos discusses innumeracy with quirky anecdotes, scenarios and facts, encouraging readers in the end to look at their world in a more quantitative way. Topics include probability and coincidence, the birthday problem, innumeracy in pseudoscience, and statistics and trade-offs in society.

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The principle is invaluable in calculating large numbers, such as the total number of telephones reachable without dialing an area code, which comes to roughly 8 x 106, or 8 million. The first position can be filled by any one of eight digits (0 and 1 aren't generally used in the first position), the second position by any one of the ten digits, and so on, up to the seventh position. (There are actually a few more constraints on the numbers and the positions they can fill, which brings the 8 million figure down somewhat.) Similarly, the number of possible license plates in a state whose plates all have two letters followed by four numbers is 262 x 104. If repetitions are not allowed, the number of possible plates is 26 x 25 x 10 x 9 x 8 x 7.

When the leaders of eight Western countries get together for the important business of a summit meeting-having their joint picture taken-there are 8x7x6x5x4x3x2x1 = 40,320 different ways in which they can be lined up. Why? Out of these 40,320 ways, in how many would President Reagan and Prime Minister Thatcher be standing next to each other? To answer this, assume that Reagan and Thatcher are placed in a large burlap bag. These seven entities (the six remaining leaders and the bag) can be lined upin7x6x5x4x 3x2x1 = 5,040 ways (invoking the multiplication principle once again). This number must then be multiplied by two since, once Reagan and Thatcher are removed from the bag, we have a choice as to which one of the two adjacently placed leaders should be placed first. There are thus 10,080 ways for the leaders to line up in which Reagan and Thatcher are standing next to each other. Hence, if the leaders were randomly lined up, the probability that these two would be standing next to each other is 10,080/ 40,320 = 1/4.

Mozart once wrote a waltz in which he specified eleven different possibilities for fourteen of the sixteen bars of the waltz and two possibilities for one of the other bars. Thus, there are 2 x 1114 variations on the waltz, only a minuscule fraction of which have ever been heard. In a similar vein, the French poet Raymond Queneau once published a book entitled Cent mille milliards de poиmes, which consisted of a sonnet on each of ten pages. The pages were cut to allow each of the fourteen lines of each sonnet to be turned separately, so that any of the ten first lines could be combined with any of the ten second lines, and so on. Queneau claimed that all the resulting 1014 sonnets made sense, although it's safe to say that the claim will never be verified.

People don't generally appreciate how large such seemingly tidy collections can be. A sports-writer once recommended in print that a baseball manager should play every possible combination of his twenty-five-member team for one game to find the nine that play best together. There are various ways to interpret this suggestion, but in all of them the number of games is so large that the players would be long dead before the games were completed.

TRIPLE-SCOOP CONES AND VON NEUMANN'S TRICK

Baskin-Robbins ice-cream parlors advertise thirty-one different flavors of ice cream. The number of possible triple-scoop cones without any repetition of flavors is therefore 31 x 30 x 29 = 26,970; any of the thirty-one flavors can be on top, any of the remaining thirty in the middle, and any of the remaining twenty-nine on the bottom. If we're not interested in how the flavors are arranged on the cone but merely in how many three-flavored cones there are, we divide 26,970 by 6, to get 4,495 cones. The reason we divide by 6 is that there are 6 = 3x2 x 1 different ways to arrange the three flavors in, say, a strawberry-vanilla-chocolate cone: SVC, SCV, VSC, VCS, CVS, and CSV. Since the same can be said for any three-flavored cone, the number of such cones is (31 x 30 x 29)/(3 x 2 x 1) = 4,495.

A less fattening example is provided by the many state lotteries which require the winner to choose six numbers out of a possible forty. If we're concerned with the order in which these six numbers are chosen, then there are (40 x 39 x 38 x 37 x 36 x 35) = 2,763,633,600 ways of choosing them. If, however, we are interested only in the six numbers as a collection (as we are in the case of the lotteries) and not in the order in which they are chosen, then we divide 2,763,633,600 by 720 to determine the number of such collections: 3,838,380. The division is necessary since there are 720 = 6 x 5x4x3x2x1 ways to arrange the six numbers in any collection.

Another example, and one of considerable importance to card players, is the number of possible five-card poker hands. There are 52 x 51 x 50 x 49 x 48 possible ways to be dealt five cards if the order of the cards dealt is relevant. Since it's not, we divide the product by (5 x 4 x 3 x 2 x 1), and find that there are 2,598,960 possible hands. Once that number is known, several useful probabilities can be computed. The chances of being dealt four aces, for example, is 48/2,598,960 (= about 1 in 50,000), since there are forty-eight possible ways of being dealt a hand with four aces corresponding to the forty-eight cards which could be the fifth card in such a hand.

Note that the form of the number obtained is the same in all three examples: (32 x 30 x 29)/ (3x2x1) different three-flavored ice-cream cones; (40 x 39 x 38 x 37 x 36 x 35)/(6 x 5 x 4 x 3 x 2 x 1) different ways to choose six numbers out of forty; and (52 x 51 x 50 x 49 x 48)/(5 x 4 x 3x2x1) different poker hands. Numbers obtained in this way are called combinatorial coefficients. They arise when we're interested in the number of ways of choosing R elements out of N elements and we're not interested in the order of the R elements chosen.

An analogue of the multiplication principle can be used to calculate probabilities. If two events are independent in the sense that the outcome of one event has no influence on the outcome of the other, then the probability that they both occur is computed by multiplying the probabilities of the individual events.

For example, the probability of obtaining two heads in two flips of a coin is 1/2 x 1/2 = 1/4 since of the four equally likely possibilities-tail,tail; tail,head; head,tail; head,head-one is a pair of heads. For the same reason, the probability of five straight coin flips resulting in heads is (1/2)5 = 1/32 since one of the thirty-two equally likely possibilities is five consecutive heads.

Since the probability that a roulette wheel will stop on red is 18/38, and since spins of a roulette wheel are independent, the probability the wheel will stop on red for five consecutive spins is (18/38)5 (or.024 -2.4%). Similarly, given that the probability that someone chosen at random was not born in July is 11/12, and given that people's birthdays are independent, the probability that none of twelve randomly selected people was born in July is (11/12)12 (or.352 – 35.2%). Independence of events is a very important notion in probability, and when it holds, the multiplication principle considerably simplifies our calculations.

One of the earliest problems in probability was suggested to the French mathematician and philosopher Pascal by the gambler Antoine Gombaud, Chevalier de Mere. De Mere wished to know which event was more likely: obtaining at least one 6 in four rolls of a single die, or obtaining at least one 12 in twenty-four rolls of a pair of dice. The multiplication principle for probabilities is sufficient to determine the answer if we remember that the probability that an event doesn't occur is equal to 1 minus the probability that it does (a 20 percent chance of rain implies an 80 percent chance of no rain).

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