CHAPTER XXXI. PROBABILITY. 1. In this chapter we shall consider the likelihood that an event, about whose happening there is uncertainty, will happen, or fail to happen. Thus, if a coin be tossed once, it may fall heads up, but it is not certain so to fall. It may fall tails up. One way of falling is as likely to happen as the other. Now, of the whole number of ways in which a coin can fall is heads up. It seems natural, therefore, to take as the mathematical expression of the likelihood, or probability, that the coin will fall heads. up. Then, is also the probability that the coin will fall tails. up. Again, let 4 white balls and 6 red balls be placed in a box, and one ball be drawn at random. If the balls cannot be distinguished by the sense of touch, one ball is as likely to be drawn as any other. Now, one ball can be drawn in 10 different cases, in 4 of which a white ball can be drawn. That is, the number of cases in which a white ball can be drawn is 16, of the whole number of cases. We therefore take as the mathematical expression of the probability of drawing at random a white ball. The probability of not drawing a white ball, which is the same as the probability of drawing a red ball, is evidently 3. 4 109 = If data relating to the number of times an event has happened in a large number of cases be collected, these data will indicate quite surely how often the same event will happen in the same number of cases under similar conditions. Thus, from tables used by life insurance companies, we find that of 95,965 healthy persons of sixteen, 95,293 have lived to We therefore take 35233 as the probability 95965 be seventeen. that a person of sixteen, in good health, will live to be seventeen. 2. The considerations of the preceding article naturally lead to the following definitions: The Favorable Cases are those in which an event can happen, or has happened in an extended number of cases. The Unfavorable Cases are those in which the event can fail to happen, or has failed to happen in an extended number of cases. The Probability that an event will happen is the ratio of the number of favorable cases to the whole number of cases. Evidently the probability that an event will not happen is the ratio of the number of unfavorable cases to the whole number of cases. If a be the number of cases in which an event can happen, and b be the number of cases in which it can fail to happen, and each case be equally likely to happen, we have: a is the probability that the event will happen; a+b b a+b is the probability that the event will not happen. The Odds in favor of an event is defined as the ratio of the number of favorable cases to the number of unfavorable cases. 3. Since an event is certain to happen or fail to happen, the number of ways favorable to its happening-or-failing is a + b. Therefore the probability of the event's happening-or-failing, that is, certainty, is 4. If P be the probability that an event will happen, it follows from the preceding article that 1-P is the probability Pis that the event will not happen. Ex. What is the probability of throwing at least 4 in a single throw with two dice? The number of cases favorable to throwing at least 4 is the number of cases in which 4, 5, 6, ..., 12 can be thrown. The number of unfavorable cases is the number of cases in which 2 and 3 can be thrown. The required probability can be obtained most readily by first finding the probability of the event's not happening. The sum 2 can be thrown in one case, 1, 1. The sum 3 can be thrown in two cases, 1, 2 and 2, 1. The two dice can be thrown in 6 × 6, = 36, different cases, counting 4, 5 and 5, 4, say, as different throws. Therefore the probability of not throwing a sum at least 4 is. 3,12; and hence the required probability is 1 — √1⁄2, = }}. = 1 127 5. Ex. A father of thirty-five has a son of twelve. What is the probability that both will be alive thirty years hence? From the table of mortality given below, we find that of 82,581 persons of thirty-five, 46,754 live to be sixty-five; that of 98,650 persons of twelve, 77,012 live to be forty-two. Now, each of the 46,754 cases favorable to the father can be taken with each of the 77,012 cases favorable to the son. That is, the number of cases favorable to both is 46,754 × 77,012. For a similar reason, the whole number of cases is 82,581 × 98,650. Therefore the required probability is 46,754×77,012 The value of this fraction to five decimal places is readily obtained by logarithms, and is .44198. Mortality Table. The following table is taken from the Actuaries' Table of Mortality, prepared from data furnished by seventeen English Life Insurance Offices. It is based on the record of 62,537 assurances, and has been generally adopted by American Companies. 1. With one die, what is the probability of throwing 6? Not 6? 6 three times in succession? 2. In a single throw with two dice, what is the probability of throwing an even number? At least 8 ? Not more than 5? 3. The letters a, e, f, r, are placed at random in a line. What is the probability that fear or fare will be written? That both vowels will come together? 4. If 52 cards be dealt to four players, what is the probability that a particular player will receive the four aces? 5. From a box containing 4 red balls, 6 black balls, and 7 white balls, 3 balls are drawn at random. What is the probability of drawing one ball of each color? 2 black and 1 white? 3 red ? 6. If 6 coins be tossed, what is the probability that they will fall 4 heads and 2 tails? 3 heads and 3 tails? 7. Nine persons are seated at random at a round table. What is the probability that A and B will be seated together? That C will be seated between A and B? 8. If 4 different volumes of history, 3 of mathematics, and 6 of literature be placed at random on a shelf, what is the probability that all the volumes in the same subject will be placed together? 9. From a box containing tickets numbered 1, 2, 3, ..., 20, three tickets are drawn at random. What is the probability of drawing 2, 3, 5? 2, 3, and not 5? Neither 2, 3, nor 5? All even numbers? Consecutive numbers ? 10-18. What are the odds in favor of the events whose probabilities are required in Exx. 1-9? Referring to the accompanying table of mortality, find the probabilities of the events in Exx. 19-21: 19. That a man of 45 will live to be 50. To be 60. To be 70. To be 80. That he will die within 5 years. years. Within 10 years. Within 20 20. That a man of 90 will live one year. Two years. Three years. Four years. Five years. At least five years. 21. At marriage, a man and his wife are 25 and 21, respectively. What is the probability that they will live to celebrate their silver wedding? Their golden wedding? 22. A representative of a firm sailed, first cabin, on a steamer which had a crew of 150 men, and which carried 150 first cabin and 250 second cabin passengers. On the voyage a man was lost. What is the probability, to the firm, that he was their representative? What, when a later report states that he was a passenger? What, when a still later report states that he was a first cabin passenger ? |