Hence, applying Theorems 3 and 1, the probability that an event will happen at least r times in n trials is P = p2 + nСn−1pn-1q + nСn-2p"-2q2 + . . . + nCrpʊqn—", which reduces to what is required. EXERCISES 1. What is the chance of making a throw with two dice that will be greater than 9? 2. A bag contains 3 dimes and 4 quarters. Three coins are drawn. Find the value of the expectation. Suggestion: Find the value of the expectation of drawing 3 quarters, 2 quarters and 1 dime, 1 quarter and 2 dimes, 3 dimes, and add. 3. Johnny may or may not receive a birthday gift of $1.00 from each of five relatives. What is the value of the expectation? 4. In a bag are five white and five black balls. If two balls are drawn what is the chance that they will both be white? Both black? one white and one black? 5. In a bag are five white and four black balls. If three balls are drawn in succession what is the chance that they will be white, black, white, in this order, if (a) the balls are not replaced each time, (b) the balls are replaced each time. 6. A man is sent 8 keys on a ring for eight locks. What is the probability that he will be able to unlock the first lock with the first key he tries? With one of the first two keys? With one of the first three? 7. Five cards are drawn from a pack of 52 cards. Find the probability that (a) there is a pair, (b) three of a kind, (c) two pairs, (d) three of a kind and a pair, (e) four of a kind, (f) a flush, (g) a straight, (h) a straight flush. 8. Three dice are thrown. What is the most probable throw? What is the probability of throwing exactly 15? At least 15? 9. If 5 letters are chosen from a group of 4 vowels and 6 consonants, what is the probability that a set will begin with a consonant and end with a vowel? 10. If there were eight independent chances in youth of growing one inch above 5 feet, find the probabilities of the statures from 5 feet to 5 feet 8 inches. 137. Mortality Tables. An important application of the theory of probability is the application to problems concerned with the duration of human life, such as life insurance, pensions, life annuities, and inheritance tax laws. Such problems are based on tables called mortality tables which show the number of deaths that may be expected to take place during a given period, among a given number of persons of a given age. These tables differ for different countries, different races, in the same country, different periods of time, and for the two sexes. Some tables are constructed from the experience of insurance companies, others from census and vital statistics reports. The American Experience Table is based on the records of the Mutual Life Insurance Company of New York. If a mortality table is based on a sufficiently large number of observations, the difference between the result furnished by the tables and actual mortality is negligible. But this must be understood to apply to large groups of people and furnishes no surety to an individual. The following table is a selection from the American Experience Table of mortality, which gives the number of people la living at age x out of 100,000 living at the age 10. The probability that a person of age x will be alive at the end of n years is denoted by Age Number living sum, the present value will be the mathematical expectation of receiving $12,220 contingent upon the probability that he will live 10 years. The probability that a person 10 years old will live at least 10 years is 120 92,637 710 100,000 10 P10 .92637. Hence the present value of the inheritance is (12,220) (.92637) = $11,320. EXERCISES 1. Plot the graph of the mortality table given in Section 137. By means of the graph estimate your own chance of living to the age of 75. 2. A man is 45 years of age and his son is 15. What is the probability that both will be alive 10 years hence? What is the probability that at least one will be alive? 3. A man and his wife are 40 and 35 years old respectively, when their child is 10 years old. What is the probability that all will be alive until the 20th anniversary of the child's birth? What is the probability that at least one will survive? 4. What should be the minimum cost of insuring the life of a person 20 years old for $1000 for five years? For insuring a couple against the death of either or both for the same sum and period if the husband is 30 and the wife 25? (Neglect interest, etc.) 5. A man makes a will leaving $40,000 to his wife in case she survives him. A son is to inherit the money if he survives both parents. If the ages of husband, wife, and son are 60, 50, 25 respectively, and if money is worth 5% compounded semi-annually, what is the present value of the expectation (a) that the wife will inherit the money in 10 years? (b) that the son will inherit the money in 15 years? 6. In each of the following exercises plot a graph with the probabilities as ordinates at arbitrary equal intervals along the x-axis. (a) A coin is tossed six times. Find the probabilities of the various ways in which it can turn up heads. (b) Find the probabilities for the various ways in which two dice can fall in one throw. (c) In the long run A wins 3 games out of 4 from B at chess. Find the probabilities of the various numbers of games which A might win in 8 successive games. (d) If a die is tossed six times, find the probabilities of the various ways in which an ace can turn up. (e) If the quantity of a trait in an individual of a group is the result of a chance combination of seven causes, determine the probabilities of the ways in which the trait may occur. 138. Frequency Distributions. At an agricultural experimental station 110 apples were classified with respect to the number of seeds each contained, and the number of apples in each class was determined. The results are given in the table. Number of Number of seeds apples 4 9 5 4 6 14 7 21 24 9 25 10 13 Thus 9 apples had 4 seeds each (the minimum number found in this investigation), 13 apples had 10 seeds each (the maximum number), while 25 apples had 9 seeds each (the most frequent number occurring). Such an arrangement of the individuals of a group, classified with respect to some characteristic which gives the number of individuals in each of the classes is called a frequency distribution and the table in which the classes and frequencies are given a frequency table. 25 A graphical representation of this analysis, called a frequency polygon, is obtained by plotting the magnitudes of the classes as abscissas, the frequencies as ordinates, and connecting the points by straight lines as in the figure. The magnitude of each class in this case is an integer and the class in Number of Apples tervals are said to vary discretely. In case the FIG. 209. characteristic measured varies continuously, as for instance the stature of a group of men, the size of the class intervals and their mid-points are chosen arbitrarily. The magnitude measured may vary discretely but by such small amounts that the number of classes is so great that the variation of the group with respect to the characteristic cannot be easily determined. In such a case the class interval is enlarged by grouping the frequencies in two or more adjacent classes and associating the resulting frequency with the midvalue of the resulting larger class. EXAMPLE. The grades in geometry of 30 students were 30, 42, 48, 55, 60, 64, 68, 71, 72, 74, 75, 76, 77, 77, 78, 78, 78, 79, 80, 82, 82, 83, 84, 85, 86, 87, 88, 88, 91, 95. Collect the data in frequency tables with class intervals of 5% and 10%. 1 95-100 Tables 1 and 2 show the data collected in class intervals of 5% with different mid-points. In table 3 the class interval is 10%. The class interval of 1% is too small for an adequate presentation of the data. Even class intervals of 5% leave some classes empty. The class interval should be chosen so as to avoid empty classes. The smaller the number of measurements the larger the class interval should be, and vice versa. The starting points. of the intervals are not so material, but it is convenient to take them so that the mid-points of the intervals are integers. In age distributions the returns usually cluster about the multiples of 5, which are taken as the mid-points of the intervals. A second method of representing frequency tables graphically is indicated in the Figs. 210-212 representing the tables above. Rectangles are constructed on the class intervals as bases with altitudes equal to the frequencies. Such diagrams are called histograms. The area of a histogram is the sum of the frequencies times the class interval. |