Probability and Expectation

The following sections are included:

• Introduction
• Contents

For convenient reference, some basic terms and concepts in the probability theory are listed below…

The first round of 2004 Europe Soccer Cup is Greece versus Portugal. At the beginning of the game, the referee let the two team captains come up and asked them to decide who should guess the result of coin tossing. Then he took out a coin and tossed it into the air. The result is the head side faced up on the ground, which is in accordance with what the Greece team captain had guessed. Therefore, the captain decided which goal his team will attack in the first half of the match. The game ended with a surprising result that Greece defeated Portugal by 2:1…

Di Qing (1008 – 1057), a distinguished military general of Northern Song Dynasty, was sent by the emperor to attack a powerful rebel army headed by Nungz Cigaoh (1025 – 1055) in South China. Before going out to fight the enemy, Di called together his troops and said: “Here are 100 copper coins, and I will toss them on the ground; if the result is that all the head sides face up, that means Heaven bless us and we will surely win the battle.” Then he tossed the coins out. What a surprise, The 100 coins were all on the ground with the head side facing up! The soldiers burst into thunderous cheers. Di also felt very happy, and said, “Let us nail these 100 copper coins on the ground now. After defeating the enemy, we will return here to celebrate our victory”…

A dice is a small cube with a different number of spots (1 to 6) on each of its sides. We assume that the dice discussed here are all made evenly, without lead or mercury put inside. So the chances that the six possible results happen are equal when rolling a dice once, i. e. all equal to …

In the 22^nd chapter of the novel The Deer and the Cauldron ( Lu Ding Ji) written by Jin Yong (Louis Cha), the hero Wei Xiao-bao had a bet with Zeng Rou, a beautiful girl disciple of the Wangwu Sect, to determine the life and death of a group of 19 persons (including Zeng). The bet would be made in the following way, Wei and Zeng roll 4 dice once, respectively, to see who get the larger sum of the spot numbers (if the sum is larger than 10 or 20, then it will be subtracted 10 or 20 before comparison)…

In a bridge card game, 52 cards are dealt evenly to four players A, B, C, D. Please find the probability that the 13 cards that player A holds is in the same suit…

When rolling a dice, how many times are needed so that the probability of getting the 1-spot at least once is greater than …

There are 4 red balls and 2 black balls in a bag, all being of the same size. Now draw one ball (not putting it back) two times. Please find the probabilities of the following events:

• the two balls drawn are both red;
• the two balls are in the same color;
• at least of one of the two balls is red.

Furthermore, if the first ball drawn is put back into the bag before the second draw, what are the answers to the three questions above?

One of the most famous story in Chinese classic novel The Three Kingdoms is that of Peach Garden Sworn Brothers: Three men — Liu, Guan and Zhang — swore to become brothers, by stating unanimously, “We are not seeking being born the same year, month, and day, but are only willing to die the same year, month, and day”…

Selecting randomly an integer from 1 to 2 000, please find the probability that neither 6 nor 8 divides the integer.…

Rolling 2 dice 4 times, we ask: what is the probability that the dice have the same spot number exactly twice…

The table tennis men's single competition in Olympic Games adopts elimination system (i.e. a player will be eliminated after losing once). That means the second-ranked player may be unfortunately eliminated before he meets the first-ranked one in the final round, despite the fact that, according to his power, he should win the silver medal…

The Knights of the Round Table in King Arthur's Court is a well-known British legend. Suppose there are 2ⁿ Knights and they will fight with each other: the losers are eliminated and the winners enter the next round of fighting. We ask: what is the probability that Balin and Balan, a pair of twin Knights, meet in the fighting? (According to the legend, the twin Knights were both killed by the other in fighting — A really bad thing! In our cases, however, the result of a fight is always one winner and one loser, no tie nor both killed. In addition, it is assumed that the win probability of each Knight is equally .)…

Three subject groups are to be formed randomly by 15 students (including 3 girls), under the condition that each group consists of 5 students and each student attends only one group. Please find the probabilities of the following events.

• (1) There is exactly one girl in each group.
• (2) The 3 girls attend the same group…

Usually the number of dice used in a game will not exceed 6. One day, however, a gambler proposed on a whim: How about playing with 12, 18 or more dice?…

There was a kingdom, where misappropriation prevailed. One day, the king sent 100 boxes, each containing 100 gold ingots, to the Chancellor of the Exchequer, who was said to be one of the most honest man in the kingdom, for preservation. The chancellor then took out a gold ingot from a box and put back a false one each day. In this way, he had stolen 100 gold ingots from the 100 boxes in 100 days. Suddenly, the king came. He took out and checked an ingot from every box. We ask, What is the probability that the king did not discover the chancellor's embezzlement?…

In a party, the host presents two pots with the same appearance, and tells you: there are two red and one black balls in one pot (named as A), and 101 red and 100 black balls in the other pot (named as B). Then he takes out a ball from one of the pots, showing it to you, and asks you whether put it back or not. After acting according to your answer, he takes out again a ball from the same pot, showing it to you, and asks you whether the pot he has drawn a ball from in the last time is A or B…

A professor has written n letters and n envelopes. His grandson puts the letters into the envelopes randomly (with each envelope containing one letter). Please find the probability that no letter matches its envelope, and the probability that there are exactly r (1 ≤ r ≤ n) matched pairs of letters and envelopes…

There are n drawers, numbered 1, 2, …, n. Now put r (≤ n) balls into these drawers. Please find the probability that each of the first r drawers (i.e. those numbered from 1 to r) contains exactly one ball…

A smoking mathematician has one box containing n matches in each of his left and right pockets. Every time he needs a match he will take out randomly a match box from one of the two pockets. Please find:

• the probability that he takes out an empty box at the first time while the other box contains exactly(r = 0, 1, …, n) matches;
• the probability that when he takes out the last match from a box, there are exactly r matches in the other one…

One scene in the Peking opera Yu Tang Chun (The Story of Su San) is called Trial in a Three-Judge Court, where three judges — the Governor Wang Jin-long, the Ponchassi Pan Bi-zheng, and the Anchasi Liu Bing-yi — interrogated jointly SuSan, an unjustly accused young woman. Wang intended to absolve Su, but Liu placed obstacles in the way. They could hardly reach an agreement on judgement…

All of the three people in a family can play chess. While it is not unusual for any of them to beat the other two in competitions, the father, however, is the best player among the three. One day, the son asked the father to buy a set of Olympic Math Tutorials for him. The latter said: “you must play chess with your mother and me in turn for three rounds; if you can win two rounds in succession, then I will buy you the books.” The son then asked: “Should I first contest with you or mother?” “You can choose it”, the father replied…

In a shooting competition at the Athens 2004 Summer Olympic Games, the American shooter Matthew Emmons was very close to winning the championship, but he accidentally cross-fired his last shot to the other player's target and finished eighth in regret…

There are five jars, among which:

• the first and second jars each contains 2 male turtles and 1 female turtle,
• the third and fourth jars each contains 3 male turtles and 1 female turtle, and
• the fifth jar contains 10 female turtles…

Let A denote the event of "test positive" in a specific experiment, and B that of "have cancer". It is known that P(B) = 0.005, P (A | B) ~ 0. 95, and . Suppose someone tests positive in the experiment. Please find the probability that he has cancer…

It is known that a particular machine will make products with a qualified rate of 90% when it is running well, but will do so with a qualified rate of only 30% when it is not running well. The probability that the machine is running well is 75% normally. Suppose that one day, the first product made by the machine is qualified. Please find the probability that the machine is running well at this time…

At the ticket window of a theater stand a queue of 2n people, each being allowed to buy one ticket with 5 yuan. Among them, n people each has only a 10-yuan note, and the other n people each has only a 5-yuan note. At the beginning, the ticket seller has no small money for change. Please find the probability that every person in the queue gets a ticket successfully, without having to wait because of the seller's unable to give him change temporarily…

Donkey and elephant are the symbols of two major political parties in the U. S. A., the Democratic and the Republican, respectively. In every election year, the politicians will have fierce competitions in seeking votes…

Lin Dai-yu, a girl heroine in the famous Chinese classic novel A Dream of Red Mansions, once said: Either East Wind overwhelms West Wind or West Wind overwhelms East Wind…

Girls from Da Ban Town is a beautiful song that has been prevailing for 60 years in China. Some words in the song say: With your dowry of millions, bring your young sister, and driving your cart, come to my house…

We continue the problem in the last chapter. Suppose the number of girls is generalized from 4 to n (≥ 4). What strategy should the suitor use?…

A company needs a secretary, and there are 10 people applying for the job. The manager of the company decides to have an interview with each of them in the time order of the applications. He will definitely deny the first three persons interviewed. Then he will deal with the following interviewees in this way: If the person during the interview shows a higher ability than all the previous interviewees, he will accept him/her; otherwise, he will deny him/her and proceed to the next interviewee. If he has denied the first 9 interviewees, he then has to accept the last one…

Boxers A and B agree to have a boxing match of an even number rounds between them. Unlike common boxing matches, this one will not sum the points of all rounds, but let every round have a winner; and the boxer who has won more than half the number of rounds will be the winner of the match…

The rule of the boxing match is the same as that in the last chapter, except that the round number here m can be either odd or even. Suppose the probability for A to win each round is . What m will make the probability for A to win the match the largest?…

Boxer A and B agree to have a match of an even number rounds between them, with the rule being the same as that in the last two chapters…

“A blind man riding a blind horse, midnight comes to the deep pool.” — This old Chinese saying is a metaphor for a very dangerous situation…

In general, suppose the horse is at the place M where it is m (a positive integer) steps away from the precipice, the probability he makes a step forward is p and that he makes a step backward is 1 − p.

Please find the probability P_m that the horse drops into the precipice…

Zhao has one yuan to gamble with and Qian has two yuan to do so. The bet in each round of the gambling between them is one yuan: the winner will take it from the loser, and there is no draw. The gamble will continue until one of them gambles away…

Continuing the problem in the last chapter, suppose Zhao and Qian each has m yuan and n yuan to gamble with, respectively, the bet for each round is one yuan, and the bet technique they possess are equal in power. Please find the probability that Zhao gambles away…

Gambler Zhao has 10 yuan, and want to earn another 10 yuan by playing roulette. It is said that the probability for a person to win in each round of roulette is 0. 474 . If he puts a yuan bet in a round, then he may either win or lose a yuan by chance after the round. He can use either an aggressive strategy — putting all the 10 yuan in one bet, or a prudent one — putting 1 yuan in each bet. Which strategy will give him the larger probability to earn 10 yuan?…

There are red and black balls of the same size in a drawer. Suppose the probability for two balls taken randomly from the drawer being both red is . At least how many balls are there in the drawer?…

A number of masters of Kongfu (martial arts), including Guo Jing, Xiaoxiang Zi, and Jinlun Fawang, are coming together to have a special competition: A small square table in the tent from a distance that can be seen by all of them. Anyone who succeeds in throwing a copper coin onto the table (the diameter of the coin is the side of the square surface of the table) and keeping the entire coin within the boundary of the table surface will win the prize…

Wang has an appointment with Zhang between 9:00 and 10:00 tomorrow morning in front of the Oriental Pearl Tower. Both are busy persons, so each can stay there for only 5 minutes. Please find the probability that they are able to meet…

Taking randomly two positive numbers x and y, both smaller than 1, suppose they form a triangle with 1. Please find the probability that the triangle is obtuse…

There are equal-distance parallel lines on a page of a usual exercise book. Suppose the distance is a. Now drop a needle of length l ( < a) onto the page randomly. Please find the probability that the needle lies across a line…

We draw at random a chord onto a circle. What is the probability that it is longer than the side of the inscribed equilateral triangle?…

Taking arbitrarily a number from the positive integers, what is the probability for it to be odd?…

Taking a number at random from interval [0, 1], what is the probability for it to be rational? And what is the probability for it to be irrational?…

Let b and c be any real numbers. Please find the probabilities that the equation

Two gamblers A and B agree that whoever wins s rounds of gambling first will take the entire stake. However, when A and B have won a and b (a, b < s) rounds, respectively, the gambling is interrupted (by unknown reason, or maybe by a police man to catch them). Please find a reasonable way to divide the stake…

A beautiful and highly gifted princess has been confined in a castle. Now there are n keys distributed to n heroes, respectively; the heroes will try to open the door of the castle with their keys in turn, and among the keys only one matches the door. We ask: Which hero has the largest probability to open the door? And how many tries in average are needed to open the door?…

Shoot a target n times in a row. It is known that the probability to hit the target each time is p. What number of hits in the n shootings can be expected?…

New Arabian Nights is a collection of short stories written by the wellknown Scottish novelist Robert Louis Stevenson (1850 – 1894), in which he told a horrible story, The Suicide Club : Every evening, the president of the club would deal cards to each member, and anyone who got the spade ace must commit to “suicide” (actually, murdered by the president)…

Suppose “the suicide club” has revised its rule, stipulating: Anyone who get the first ace (no matter which suit it belongs to) must commit to suicide. How many cards in average will be dealt before the first ace appears?…

There are two packs of poker cards. We at first place the first pack of cards (52 in all) in a row, from left to right, and then place the second pack on the first ones, respectively, to form 52 card couples. How many pairs (i. e. those consisting of the same card) are there in average among these card couples?…

Republic of Roska practices welfarism, stipulating: A factory should give all his workers the entire day off, as long as it is the birthday of any worker in the factory. Therefore, if a factory has only one worker, then there are in fact 364 man-days in a year; if it has two worker, there may be 2 × 363 man-days yearly. In order to expect the largest number of man-days in a year, how many workers should a factory have? (Assume that there are 365 days in a year.)…

You buy a lottery ticket with one yuan, and then have a chance to win 100 000 yuan. The large reward is attracting a large number of lottery consumers. However, you only have a one-millionth chance to win the lottery. Please find the expected value of a lottery ticket…

A gambler is allowed to select a number from 1, 2, 3, 4, 5, 6, and stake a yuan on it. Now, rolling three dice, if the selected number appears once, twice or three times, then the gambler will get money one, two or three times of the stake, respectively, and the original stake will also be returned to him. However, if the selected number does not appear, he will lose the stake…

There are 8 girls and 7 boys in a social party, sitting around a round table. If all the girls sit together, there are then only two girls adjacent to boys. If girls and boys sit as alternately as possible, then there are 14 pairs of seats that are girl and boy adjacent. How many pairs of seats are there in average that are girl and boy adjacent?…

Young man Wang takes a dice and wants to roll out the largest spot number — the 6-spot. But the result is not so. (We know that the probability to roll out the 6-spot with a dice is , and that to get a contrary result is )…

A cigarette factory starts a promotion activity: Each pack of cigarettes will contain a picture of one of the 108 heroes from the Chinese traditional novel The Water Margin. If a consumer is able to collect all the 108 heroes by buying enough cigarettes produced by this factory, he will be given a reward. Please find the average number of packs of cigarettes the consumer should buy in order to win the reward…

An addicted gambler has a psychologist friend. The latter hopes to help the former in giving up gambling, and then offers a bet to him: Let the gambler play a gamble for 36 rounds. If he cannot win even once, then he should pay the psychologist 100 yuan and will gamble no more; otherwise, the psychologist will pay him 100 yuan. It is known that the bet for each round is 1 yuan — If the gambler wins, he can get 30 yuan as well as back the bet paid; otherwise, he will lose the bet…

An evenly made glass rod with length l has fallen and broken into two fragments, One longer and the other shorter. We ask: How long in average is the longer fragment, as well as that of the shorter one?…

Suppose the glass rod discussed in the last chapter has been broken into three fragments. How long averagely are the shortest, the longest, and the middle length fragments, respectively?…

Suppose the side of each square on a squared paper is 1. Now drop a needle with length l on the paper. Please find the average number of points at which the needle and lines meet…

There are equal-distance parallel lines on a flat plane. Suppose the distance is a. Now throw a wire ring with circumference 2l (2l < πa) onto the plane. Please find the probability for the ring lying across lines…