INTRODUCTION TO PROBABILITY. Mathematics in Daily Life

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1 INTRODUCTION TO PROBABILITY Mathematics in Daily Life Some shopping centers invite customers to try their luck by joining lucky draw or playing a spinner after buying certain amounts of products. Customers will win the prize that the pointer points to after the spinner stops spinning. It is a matter of luck and chance called probability. We can use numbers (called probability) to show how likely a customer is to win a particular prize. Introduction to Probability In our daily lives, we often guess whether a situation will happen or not. We forecast the chance of the occurrence of an event. In mathematics, an event is a situation that may happen. We can study the following examples of events. From the above sentences, we recognize that the terms certain, very likely, unlikely and impossible can be used to state how likely an event will happen. However, these terms cannot tell the extent of certainty. In mathematics, we use a number, which is known as the probability, to measure how likely it is that an event will occur. The larger the number is, the greater is the chance of an event occurring. Each result of the

2 event that is likely to happen, is called a possible outcome. For example, there are possible outcomes of tossing a fair coin once: a head and a tail. Probability of an Event The probability of an event is the chance of the occurrence of that event. We define the probability of an event as the ratio of the number of favorable outcomes of the event to the total number of possible outcomes. The probability of an event E is denoted by P(E). P E Number of favorable outcomes of an event Number of possible outcomes E Example : A letter is selected randomly from the word PROBABILITY. Find the probability of getting (a) a vowel, (b) a B or a T Solution: (a) There are vowels in the word: O, A and I s P(getting a vowel) = (There are letters in the word PROBABILITY ). (b) There are B s and T in the word. P(getting a B or T ) = 3 Example : A box contains red balls and yellow balls. If a ball is randomly chosen from the box, find the probability that (a) it is a red ball, (b) it is a yellow ball, (c) it is a black ball, (d) it is a colored ball. Solution: (a) P(getting a red ball) = (Number of red balls) (Total number of balls = Number of red balls + Number of yellow balls) = 6 = 3 (b) P(getting a yellow ball) = 6 = 3 (c) P(getting a black ball) = 0 6 = 0 (d) P(getting a colored ball) = 6 = 6 6 = Exercises. There are options for a multiple-choice question and only one of them is correct. If a student guesses the answer randomly, find the probability of getting (a) the correct answer (b) a wrong answer. A letter is selected randomly from the word UNKNOWN. Find the probability of getting (a) a vowel (b) a N or a K 3. An integer is selected at random from to 5 inclusively. Find the probability of getting an odd number. Find the probability of getting an odd number.

3 . There are $-coins and 5 $-coins in rosa s purse. If she draws a coin from her at random, find the probability that it is a (a) $-coin (b) $ -coin (c) $5-coin 5. There are 0 novels and comics on a bookshelf. Tony chooses a book from the bookshelf at random. Find the probability of choosing (a) a novel (b) a comic (c) a comic or a novel 6. A fair die is thrown. Find the probability of getting (a) an odd number (b) a number smaller than 3 (c) a number greater than 6 7. John forgets the last digit of tom s telephone number. If john randomly dials a number for the last digit, what is the probability that he can actually call tom? 8. If a student is chosen from a school at random, find the probability that the student was born (a) in November (b) on Tuesday Answer:. (a) (b) 3. (a) 7 (b) (a) 9 (b) (a) 5 8 (b) 3 8 (c) 6. (a) (b) 3 (c) (a) (b) 7

4 Methods for Listing Possible Outcomes According to the method learn in the previous, we need to know the total number of all the possible outcomes before calculating the probability of an event. When the situation is complicated, we have to use some methods for listing all the possible outcomes to avoid missing or repeating data. Tree Diagram A restaurant offers three types of set dinners: beef, pork and mutton, and each meat dish can be combined with either vegetable soup, cream soup or onion soup. If we want to list out all the possible combinations systematically, we can use symbols, single words or letters to represent a selection. For example, we use A, B and C respectively to stand for beef, pork and mutton; X, Y and Z respectively to stand for vegetable soup, cream soup and onion soup. Finally, the following tree diagram can be drawn. Set Dinner Soup Combination A X AX Y AY Z AZ B X BX Y BY Z BZ C X CX Y CY Z CZ From the tree diagram above, we can see that there are 9 possible combinations. Example Sophie has bought two VCD movies and two DVD movies. She chooses two of them randomly and lends them to William. Find the probability of each of the following events. a. William has borrowed to DVDs b. William has borrowed one VCD and one DVD. Solution: Let V, V stand for the two VCDs and D, D stand for the two DVDs. The tree diagram below shows all the possible outcomes of borrowing two discs from Sophie: st Disc nd Disc Outcome V V V V D V D D V D V V V V D V D D V D D V D V V D V D D D D V D V V D V D D D

5 NO. ON THE ST DICE st Disc Two number of possible outcomes = a. Number of favorable outcomes = (The favorable outcomes are D D and D D )3 P(two DVDs) = = 6 b. Number of favorable outcomes = 8 (The favorable outcomes are V D, V D, V D, V D, D V, D V, D V, D V ) P(one VCD and one DVD) = 8 = 3 Tabulation Referring to Example on the previous page, if Sophie has bought one more DVD movie, then the total number of possible outcomes for William borrowing discs will increase greatly, and a tree diagram will become complicated. Under this situation, it may be better to use a table to list out all possible outcomes. nd Disc V V D D D 3 V X V V V D V D V D 3 V V V X V D V D V D 3 D D V D V X D D D D 3 D D V D V D D X D D 3 D 3 D 3 V D V D 3 D D 3 D X Table Using the above table, we have P(two VCDs) = 0 = 0, P(only one DVD) = 0 = 3 5. Example Two fair dice are thrown. Find the probability of each of the following events. a. The numbers on both dice are prime numbers. b. The sum of the numbers on the two dice is 8. c. At least one dice shows up a 3. Solution: No. on the nd dice 3 5 6,,, 3,, 5, 6,,, 3,, 5, 6 3 3, 3, 3, 3 3, 3, 5 3, 6,,, 3,, 5, 6 5 5, 5, 5, 3 5, 5, 5 5, 6 6 6, 6, 6, 3 6, 6, 5 6, 6 Table Total number of possible outcomes = 6 6 = 36 a. Number of favorable outcomes 9 (The favorable outcomes are (, ), (, 3), (, 5), (3, ), (3, 3), (3, 5), (5, ), (5, 3) and (5, 5)) P(both prime) = 9 36 = b. Number of favorable outcomes = 5 (The favorable outcomes are (, 6), (3, 5), (, ), (5, 3) and (6, )) P(total is 8) = 5 36 c. Number of favorable outcomes = (The favorable outcomes are (, 3), (, 3), (3, ), (3, ), (3, 3), (3, ), (3, 5), (3, 6), (, 3), (5, 3) and (6, 3)) P(at least one dice shows up a 3) = 36

6 Geometric Probability In the last section, we have studied the method of counting all the possible outcomes in order to find the probability of an event. However, under some situations, it is impossible for us to use the method of counting the outcomes. Let us consider the following question: In the figure below. AB is a line segment 5 cm in length. A point X on AB is selected at random, what is the probability that the length of AX is longer than 3 cm? A B 5 cm In order to answer this question, we can let C be a point on the line segment AB such that it is 3 cm from A [see figure below]. To make AX > 3 cm, the position of X must be within CB. Therefore, P(AX > 3 cm) = P(X is within CB) Obviously, the probability that X is within a certain part of AB should be proportional to the length of that part, therefore lenght of CB P X is within CB lenght of AB 5 P AX 3cm 5 This kind of problems involve calculating probabilities by considering certain measures of geometric figures such as lengths, areas, volumes, etc. They are related to the concept of geometric probability. Its basic principle is as follows: In a geometric figure, if an event E happens in a certain region of the figure, then measure of the region in which E happens PE same measure of the whole figure Example 3: In a stall game, a participant has to throw a dart at the circular target as shown in the figure. A big prize will be awarded if the red bullseye is hit. Teddy participates in the game. He throws a dart at random and it hits the target. Find the probability that the dart hits the bulls eye. Solution: area of bulseye Phitting the bullseye area of target 5 cm 0 cm 6 Exercise. Janice has children. a. List all the possible outcomes of gender of the children using a tree diagram. b. Find the probability that both children are boys.. Ken throws 3 fair coins. a. List all the possible outcomes using a tree diagram. b. Find the probability that Ken gets at least head. A 5 cm cm C 3. A set lunch includes a soup and a sandwich. The restaurant offers the following choices. B

7 a. List all the possible set lunches using a tree diagram. b. Mrs. Chan randomly orders a set lunch but she dislikes chicken and salmon. Find the probability that the set lunch ordered suits her friend s taste?. letters are chosen randomly from the word BOOK. a. List the possible outcomes using a table. b. Find the probability that vowels are selected. 5. There are cans of cola and cans of orange juice in a refrigerator. Harry takes cans of drink at random. a. List the possible outcomes using a table. b. Find the probability that one of each kind of drink is selected. 6. The figure shows a dartboard with 6 identical equilateral triangles. Raymond throws a dart at the dartboard. If the dart hits the orange region, he will win the grand prize. If the dart hits the purple regions, he will win the nd prize. He will not get a prize if he hits the other regions. a. Find the probability that Raymond wins the second prize. b. Find the probability that Raymond does not get a prize. 7. The figure shows the top view of a mango cake with outermost edge decorated with mouse of width cm. A fly lands on the cake. Find the probability that the fly lands on the mouse. Solution for Exercises:. b.. b. 3. b.. b. 5. b. 6. b. b