A History of Mathematics: An Introduction

Victor J.Katz

Chapter 18 Probability and Statistics in the Eighteenth Century - all with Video Answers

Educators

Chapter Questions

Problem 1

Calculate the Bernoulli numbers $B_{8}, B_{10}$, and $B_{12} .$ The sequence of Bernoulli numbers is usually completed by setting $B_{0}=1, B_{1}=-\frac{1}{2}$, and $B_{k}=0$ for $k$ odd and greater than 1 .

Foster Wisusik

Numerade Educator

01:09

Problem 2

Write out explicitly, using Bernoulli's techniques, the formulas for the sums of the first $n$ fourth, fifth, and tenth powers. Then show that the sum of the tenth powers of the first 1000 positive integers is
$$
91,409,924,241,424,243,424,241,924,242,500
$$
Bernoulli claimed that he calculated this value in "less than half of a quarter of an hour" (without a calculator).

Tanishq Gupta

Numerade Educator

01:16

Problem 3

Show that if one defines the Bernoulli numbers $B_{i}$ by setting
$$
\frac{x}{e^{x}-1}=\sum_{i=0}^{\infty} \frac{B_{i}}{i !} x^{i}
$$
then the values of $B_{i}$ for $i=2,4,6,8,10,12$ are the same as those calculated in the text and in Exercise 1 .

Hoan Nguyen

Numerade Educator

02:35

Problem 4

Suppose that $a$ is the probability of success in an experiment and $b=1-a$ is the probability of failure. If the experiment is repeated three times, show that the probabilities of the number of successes $S$ being $3,2,1,0$, respectively, are given by $P(S=3)=1 a^{3}, P(S=2)=3 a^{2} b, P(S=1)=$ $3 a b^{2}$, and $P(S=0)=1 b^{3}$.

Prabhakar Kumar

Numerade Educator

02:26

Problem 5

Generalize Exercise 4 to the case of $n$ trials. Show that the probability of $r$ successes is $P(S=r)=\left(\begin{array}{c}n \\ n-r\end{array}\right) a^{r} b^{n-r}$.

Aishwarya Krishnakumar

Numerade Educator

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Problem 6

Using the results of Exercise 5 , with $a=1 / 3, b=2 / 3$, and $n=10$, calculate $P(4 \leq S \leq 6)$.

Victor Salazar

Numerade Educator

01:14

Problem 7

Complete Bernoulli's calculation of his example for the Law of Large Numbers by showing that if $r=30$ and $s=20$ (so $t=50$ ) and if $c=1000$, then
$$
n t+\frac{r t(n-1)}{s+1}>m t+\frac{s t(m-1)}{r+1}
$$
where $m, n$ are integers such that
$$
m \geq \frac{\log c(s-1)}{\log (r+1)-\log r}
$$
and
$$
n \geq \frac{\log c(r-1)}{\log (s+1)-\log s}.
$$
Conclude that in this case the necessary number of trials is $N=25,550$.

Hoan Nguyen

Numerade Educator

05:14

Problem 8

Use Bernoulli's formula to show that if greater certainty is wanted in the problem of Exercise 7 , say, $c=10,000$, then the number of trials necessary is $N=31,258$.

Ahmed Genedy

Numerade Educator

01:30

Problem 9

In his Letter to a Friend on Sets in Court Tennis, written in 1687 but not published until 1713, Jakob Bernoulli analyzed the probabilities at any point in a game or set of court tennis, whose scoring rules are virtually identical with those of tennis today. He determined the odds both when the players were evenly matched and when one player was stronger than the other. If two players $A$ and $B$ are evenly matched in a tennis game with the score $15: 30$, determine the probability of player $A$ winning. (Remember that one must win by two points.)

Nick Johnson

Numerade Educator

04:19

Problem 10

Continuing from Exercise 9, suppose that player $A$ is twice as strong as player $B$. Suppose that the score is $30: 30$. Determine the probability of player $A$ winning. What is the probability of $A$ winning if the score is $15: 30$ ?

Adriano Chikande

Numerade Educator

00:33

Problem 11

Suppose that the probability of success in an experiment is $1 / 10$. How many trials of the experiment are necessary to ensure even odds on it happening at least once? Calculate this both by De Moivre's exact method and his approximation.

Hossam Mohamed

Numerade Educator

00:41

Problem 12

How many throws of three dice are necessary to ensure even odds that three ones will occur at least once?

Trinity Steen

Numerade Educator

00:36

Problem 13

In a lottery in which the ratio of the number of losing tickets to the number of winning tickets is $39: 1$, how many tickets should one buy to give oneself even odds of winning a prize?

Sneha Ravi

Numerade Educator

03:12

Problem 14

Generalize De Moivre's procedure in Problem III (of his text) to solve Problem IV: To find how many trials are necessary to make it equally probable that an event will happen twice, supposing that $a$ is the number of chances for its happening in any one trial and $b$ the number of chances for its failing. (Hint: Note that $b^{x}+x a b^{x-1}$ is the number of chances in which the event may succeed no more than once, while $(a+b)^{x}$ is the total number of chances.) Approximate the solution for the case where $a: b=1: q$, with $q$ large, and show that $x \approx 1.678 q$.

Lourence Gonhovi

Numerade Educator

01:57

Problem 15

Show that the sums labeled col. 1, col. 2, and col. 3 in De Moivre's derivation of the ratio $\left(\begin{array}{c}n \\ n / 2\end{array}\right): 2^{n}$ may be written explicitly as
col. $1=\frac{s^{2}+s}{m}$
$\operatorname{col} 2=\frac{\frac{1}{2} s^{4}+s^{3}+\frac{1}{2} s^{2}}{3 m^{3}}$
$\operatorname{col} 3=\frac{\frac{1}{3} s^{6}+s^{5}+\frac{5}{6} s^{4}-\frac{1}{6} s^{2}}{5 m^{5}}$
Determine the corresponding value for col. $4 .$

Henry Mak

Numerade Educator

01:04

Problem 16

Add the highest-degree terms of the columns from Exercise 15 to get
$$
s\left(\frac{s}{m}+\frac{1}{2 \cdot 3} \frac{s^{3}}{m^{3}}+\frac{1}{3 \cdot 5} \frac{s^{5}}{m^{5}}+\frac{1}{4 \cdot 7} \frac{s^{7}}{m^{7}}+\cdots\right)
$$
which, setting $x=s / m$, is equal to
$$
s\left(\frac{2 x}{1 \cdot 2}+\frac{2 x^{3}}{3 \cdot 4}+\frac{2 x^{5}}{5 \cdot 6}+\frac{2 x^{7}}{7 \cdot 8}+\cdots\right)
$$
Show that the series in the parenthesis can be expressed in finite terms as
$$
\log \left(\frac{1+x}{1-x}\right)+\frac{1}{x} \log \left(1-x^{2}\right)
$$
and therefore that the original series is
$$
m x \log \left(\frac{1+x}{1-x}\right)+m \log \left(1-x^{2}\right)
$$
Since $s=m-1$ (or $m x=m-1$ ), show therefore that the sum of the highest-degree terms of the columns of Exercise 15 is equal to
$$
\begin{aligned}
&(m-1) \log \left(\frac{1+\frac{m-1}{m}}{1-\frac{m-1}{m}}\right) \\
&\quad+m \log \left[\left(1+\frac{m-1}{m}\right)\left(1-\frac{m-1}{m}\right)\right]
\end{aligned}
$$
which in turn is equal to $(2 m-1) \log (2 m-1)-2 m \log m$.

Mir Afzal

Numerade Educator

00:36

Problem 17

Show that the sum of the second-highest-degree terms of each column from Exercise 15 is
$$
\frac{s}{m}+\frac{s^{3}}{3 m^{3}}+\frac{s^{5}}{5 m^{5}}+\frac{s^{7}}{7 m^{7}}+\cdots,
$$
which, since $s=m-1$, is equal to
$$
\frac{1}{2} \log \left(\frac{1+\frac{s}{m}}{1-\frac{s}{m}}\right) \text { or } \frac{1}{2} \log (2 m-1).
$$

Amy Jiang

Numerade Educator

00:40

Problem 18

Derive De Moivre's result
$\log \left(\frac{Q}{M}\right) \approx-\frac{2 t^{2}}{n} \quad$ or equivalently $\quad \log \left(\frac{M}{Q}\right) \approx \frac{2 t^{2}}{n}$
(Hint: Divide the arguments of the first two logarithm terms in the expression in the text by $m$. Then simplify and replace the remaining logarithm terms by the first two terms of their respective power series.)

Linh Vu

Numerade Educator

01:06

Problem 19

De Moivre's result developing the normal curve implies that the probability $P_{\epsilon}$ of an observed result lying between $p-\epsilon$ and $p+\epsilon$ in $n$ trials is given by
$$
P_{\epsilon}=\frac{1}{\sqrt{2 \pi n p(1-p)}} \int_{-n \epsilon}^{n \epsilon} e^{-\frac{t^{2}}{2 n p(1-p)}} d t
$$
Change variables by setting $u=t / \sqrt{n p(1-p)}$ and use symmetry to show that this integral may be rewritten as
$$
P_{\epsilon}=\frac{2}{\sqrt{2 \pi}} \int_{0}^{\frac{\sqrt{n} \epsilon}{\sqrt{p(1-p)}}} e^{-\frac{1}{2} u^{2}} d u
$$
Calculate this integral for Bernoulli's example, using $p=$ $.6, \epsilon=.02$, and $n=6498$, and show that in this case $P_{\epsilon}=$ $0.999$, a value giving moral certainty. (Use a graphing utility.) Find a value for $n$ that gives $P_{\epsilon}=0.99$.

Manik Pulyani

Numerade Educator

01:13

Problem 20

Calculate $P(r<x<s \mid X=n-1)$ explicitly, using Bayes's theorem. In particular, suppose that you have drawn 10 white and 1 black ball from an urn containing an unknown. proportion of white to black balls. If you now guess that this unknown proportion is greater than $7 / 10$, what is the probability that your guess is correct?

Gregory Higby

Numerade Educator

02:30

Problem 21

Show that if an event of unknown probability happens $n$ times in succession, the odds are $2^{n+1}-1$ to 1 for more than an even chance of its happening again.

Mengchun Cai

Numerade Educator

02:25

Problem 22

Imagine an urn with two balls, each of which may be either white or black. One of these balls is drawn and is put back before a new one is drawn. Suppose that in the first two draws white balls have been drawn. What is the probability of drawing a white ball on the third draw?

Narayan Hari

Numerade Educator

03:31

Problem 23

With interest at $4 \%$, what is the present value of an annuity of one pound per year for 50 years?

Charles Carter

Numerade Educator

01:07

Problem 24

With interest at $4 \%$, what is the present value of a life annuity of one pound per year for someone of age 36 ?

Linh Vu

Numerade Educator

01:48

Problem 25

In the French Royal Lottery of the late eighteenth century, five numbered balls were drawn at random from a set of 90 balls. Originally, a player could buy a ticket on any one number or on a pair or on a triple. Later on, one was permitted to bet on a set of four or five as well as on a set given in the order drawn. Show that the odds against winning with a bet on a single number, a pair, and a triple are $17: 1,399.5: 1$, and $11,747: 1$, respectively. The payoffs on these bets are 15,270 , and 5,500 .

Manisha Sarker

Numerade Educator

01:19

Problem 26

In Euler's analysis of the lottery for the case $k=2$, determine the general formulas for the "fair" prizes $a$ and $b$ for matching two numbers and for matching one number, respectively, in terms of $n$ and $t$, where $t$ tokens are drawn out of a total of $n$.

James Chok

Numerade Educator

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Problem 27

Work out the probabilities for the case $k=3$ in the lottery described in the text. That is, assuming that $t$ tokens are drawn out of a total of $n$, find the probabilities that if a player picks three numbers, he will match all three, match two, or match one of the numbers drawn.

Donna Densmore

Numerade Educator

01:10

Problem 28

For the case of the lottery from Exercise 27 , determine the specific probabilities that a player will match three numbers, match two numbers, or match one number in the case $n=90$ and $t=5$.

Harsh Gadhiya

Numerade Educator

04:01

Problem 29

In the situation of Exercise 28 , find the "advantage" of the player and determine the equation that determines "fair" prizes $a, b$, and $c$ for each of the three possibilities, assuming a bet of 1 ecu. (Here $a$ is the prize for matching all three numbers, $b$ the prize for matching two numbers, and $c$ the prize for matching one number.)

Suman Saurav Thakur

Numerade Educator

02:43

Problem 30

In the situation of Exercise 29 , assume that the three summands in the equation are each equal to $1 / 3$. Determine the prizes in that event. Then assume that the first summand (for matching all three numbers) is $1 / 7$, while the other two are each 3/7, and determine the prizes. Finally, assume that the first summand is $1 / 16$, the second (for matching two numbers) is $6 / 16$, and the third (for matching one number) is $9 / 16$ and determine the prizes. (These three cases are all discussed by Euler.)

Debasish Das

Numerade Educator

00:38

Problem 31

The so-called St. Petersburg Paradox was a topic of debate among those mathematicians involved in probability theory in the eighteenth century. The paradox involves the following game between two players. Player $A$ flips a coin until a tail appears. If it appears on his first flip, player $B$ pays him 1 ruble. If it appears on the second flip, $B$ pays 2 rubles, on the third, 4 rubles, $\ldots$, on the $n$th flip, $2^{n-1}$ rubles. What amount should $A$ be willing to pay $B$ for the privilege of playing? Show first that $A$ 's expectation, namely, the sum of the probabilities for each possible outcome of the game multiplied by the payoff for each outcome, is
$$
\sum_{i=0}^{\infty} \frac{1}{2^{i}} 2^{i-1}
$$
and then that this sum is infinite. Next, play the game 10 times and calculate the average payoff. What would you be willing to pay to play? Why does the concept of expectation seem to break down in this instance?

Hunza Gilgit

Numerade Educator

01:05

Problem 32

Outline a lesson for a statistics course deriving Bayes's theorem and discussing its usefulness.

Lauren Shelton

Numerade Educator