In China, lots of excellent students who are good at maths take an active part in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results — they have won the first place almost every year.
The author is one of the senior coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.
This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. This book will, in an interesting problem-solving way, explain what probability theory is: its concepts, methods and meanings; particularly, two important concepts — probability and mathematical expectation (briefly expectation) — are emphasized. It consists of 65 problems, appended by 107 exercises and their answers.
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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.
Assuming that the coin is made evenly, therefore, when it reaches the ground, the chance of the head side up and that of the tail side up are the same, i.e. half to half, or both is equal to
.
The “chance” mentioned above is called “probability” in mathematics.
Suppose there are n possible results in an experiment, and the occurrence probability of each result is the same; if among them there are m results belonging to event A, then we say the probability of A is
, expressed as P(A) =
, where P(A) denotes the probability of A.
For example, when tossing a coin, there are two possible results: the head side up (denoted by A) and the tail side up (denoted by B). So n = 2, and
Now tossing a coin three times, please find
(1) the probability that the head side occurs just once, and
(2) the probability that the head side occurs at least once.
Solution: (1) When tossing a coin three times, there are 8 possible results, as shown below,
(H = the head side up, and T = the tail side up). Among them there are 3 results belonging to the event “the head side occurs just once”, so the required probability is
.
In general, tossing a coin n times will produce 2n possible results, among which there are
results belonging to the event “the head side up occurs k times exactly”. So the occurrence probability of this event is
(2) Among the 8 possible results mentioned above, there is only one in which the head side does not face up even once in the three coin tosses, while in the other 7 results the head side faces up at least once. So the probability required is
.
In general, when tossing a coin n times, the probability that the head side does not face up even once is
and that the head side faces up at least once
When we say that, in tossing a coin, the probabilities of the head side up and the tail side up are both equal to
, we do not mean that the two events will both occur
time in a toss. The number of times must be a nonnegative integer. So in every time of tossing a coin, the result is either the head side up or the tail side up, and will never be half time the head side up and half time the tail side up. If tossing a coin many times, however, the number of the head side up and that of the tail side up are roughly equal. Although this conclusion seems apparent, some conscientious people still made efforts to test it. For example, the famous French scholar Georges-Louis de Buffon (1707 – 1788) had tossed a coin more than 4 thousand times, and obtained the result as shown below:
Total number of tests
The head side up
Frequency
4040
2048
0.5069
Here, the frequency (of the head side up) is the ratio between the number of the head side up and that of the total tosses, i.e.
Another scholar, the great English statistician Karl Pearson (1857 – 1936), went even further by doing the test 2 times, and obtained the result as shown below:
From the table above we see that the chance that the head side faces up is really about
, and its frequency is tending to the probability
with the increase of the number of tosses.
2.General Di Qing’s Coins
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...
