Worksheet for Teaching Module Probability (Lesson 1) Topic: Basic Concepts and Definitions Equipment needed for each student 1 computer with internet connection Introduction In the regular lectures in school, some introductory concepts of probability are taught. We are told that Number of favourable outcomes Probability =. Number of possible outcomes This definition of probability usually gives us a correct sense of the chance of an event to happen usually. However, sometimes it is not so. In this lesson, we are going to find a more reasonable definition of probability. Some misconceptions will also be discussed. Activity 1 (Reading Comics) Read the comics in the file comics_1.pdf and answer the following questions. 1. If they play the game, do they have any chance to win? 2. Consider the distribution of the powers of 10 among all positive integers. Do you agree that the probability to win is 0 [Note 1]? If you answer yes in Question 2, go to Question 3. Otherwise, jump to Question 4. 1
3. Is there any contradiction between your answers in the first 2 questions? Can you explain it? 4. It is known that we can order positive integers as 1, 2, 10, 3, 100, 4, 1000, 5,. Do you agree with William that this ordering gives another probability 1 2? 5. How about this ordering: 1, 2, 3, 10, 4, 5, 100, 6, 7, 1000, 8, 9,? Does it imply the probability to win is 1? Is the result in this question consistent with the results in Questions 2 3 and 4? 6. Recall the definition of probability taught in school. Can you calculate the probability with this definition? You should be confused with the example in the comics, right? Actually, this confusion arises from the definition. Let s try a game again to derive a better definition. Activity 2 (Roulette) Recall that the theoretical probability of an event can be estimated by repeating an experiment many times. In this activity, we are going to estimate the winning probability of the game shown in the comics. Go to the webpage http://www.mathdb.org/, click into Teaching Modules > Probability > Lesson 1 > Activity 2 and run the program Roulette. It simulates a game of a roulette wheel. On the wheel some integers are written, and if you play the wheel and get a power of 10, then you win. 2
First of all, play the wheel with 10 regions 100 times. Record the results in the table: Number of trials: Number 1 10 Number of appearances Experimental Probability 7. What is the experimental probability to win (i.e. getting a power of 10)? 8. Please use the formula stated in the introduction to compute the theoretical probability to win. Do the experimental result and your calculated result match? Next, play the wheel with 100 regions 1000 times. Record the results in the table: Number of trials: Number 1 10 100 Number of appearances Experimental Probability 9. What is the experimental probability to win now (i.e. getting a power of 10)? 10. Compute the theoretical probability to win as in Question 8. Do the experimental result and your calculated result match? There is another wheel with 1000 regions on which 1 to 1000 are written. 11. Before trying this wheel, please calculate the theoretical probability first. 3
Play the wheel 10000 times and record the results in the table. Number of trials: Number 1 10 100 1000 Number of appearances Experimental Probability 12. What is the experimental probability to win? 13. Do the experimental result and your calculated result match? Activity 3 (Discussions) 14. Can you think of the reason for the coincidence of the experimental and calculated results in the first 2 wheels? 15. What is the assumption needed for the formula stated in the introduction? 16. Is the assumption ALWAYS valid? Here is another story: Candy was going to have an exam the next day. However, she wanted to play a new computer game after school. She decided to leave her books behind according to her theory: The exam will be cancelled if a rainstorm signal is issued tomorrow. There are 3 rainstorm signals, namely black, red and amber. Therefore, exactly one of 4 possibilities will happen tomorrow: Black, red, amber or no signal at all. 3 out of 4 possible outcomes are favourable, so there is a probability 4
of 3 4 that I do not need to go to school. The probability is so high. I should be lucky enough to escape from the exam. Finally, it was sunny the next day, and of course, she failed in the exam. 17. Is the probability really 3? Did she make any unreasonable assumption in her calculation? 4 18. After reading all the examples, can you think of a better way to define probability to overcome this problem? (Please relax if you don t come up with any idea. In that case you may simply skip this and the next question.) In probability theory, we define probability as follows. Let S be a non-empty set; we call it sample space. By the word event we mean a subset of S [Note 2]. Denote the collection of all events by E. A function P : E [0,1] is said to be a probability function (or simply probability) on S if i. P( S ) = 1; and ii. P( E E ) P( E ) P( E ) 1 2 = 1 + 2 + whenever E 1, E 2,... is a sequence of mutually disjoint events. (By mutually disjoint we mean Ei Ej = whenever i j.) The value P( E ) is called the probability of the event E. 19. Let s recall the game in the comics. If we define the probability with this definition, can we still have the assumption discussed in Question 15 (namely, all outcomes have equal chance to occur)? If not, what do we need in order to find the probability to win? 5
Conclusion (A Better Definition of Probability) The probability of an event does not come naturally with the sample space. It needs to be defined separately. It is not a must for all outcomes to have the same probability. The definition taught in school is insufficient to deal with all cases. If not all of the outcomes have the same probability to occur, the formula may fail. Suppose a number is taken randomly from the set {1, 2,..., n}. What is the probability of getting 1? Some people may answer 1/n, because he/she implicitly assumes that all numbers come up with the same probability. In the case where the sample space is finite, there is a natural probability function defined on it the uniform distribution. However, it is certainly not the case when the sample space is infinite. Questions such as what is the probability of getting a power 10 if a positive integer is taken randomly? are in fact NOT well-defined, unless the probability function is given. Note: 1. An event with zero probability does not mean that there is no way for it to occur. (In mathematical terms, the event need not be empty.) 2. Indeed, there are some technical restrictions on the collection E of all events. In general a subset of the sample space S need not be an event. However, throughout this module E is taken to be the power set of S so that all subsets of S are events. ~ End of worksheet ~ 6