Learn to find the theoretical probability of an event.

Theoretical probability is used to find the probability of an event when all outcomes are equally likely. Equally likely outcomes have the same probability.

If each possible outcome of an experiment is equally likely, then the experiment is said to be fair. Experiments involving number cubes and coins are usually assumed to be fair.

Example A In the game of Bingo, balls containing a combination of a letter and a number are randomly selected. Players try to match the combinations on 5 x 5 grid cards. Of the 75 combinations used in Bingo, 15 have the letter B on them. What is the probability of selecting one of the 15 Bs from 75 total Bingo Balls? See Textbook Pg. 418 for solution.

Example B What is the probability of rolling a number greater than 2 on a fair number cube? See Textbook Pg. 419 for solution.

Example C There are 11 boys and 16 girls in Mr. Ashley’s class. Mr. Ashley has written the name of each student on a craft stick. He randomly chooses one of these sticks to choose a student to answer a question. What is the probability that the student is a boy? What is the probability that the student is a girl? See Textbook Pg. 419 for solution.

YOU TRY 1a P = P(clear) = = = 0.45 = 45% Andy has 20 marbles in a bag. Of these, 9 are clear and 11 are blue. Find the probability of drawing a clear marble from the bag. number of ways the event can occur total number of equally likely outcomes P = number of clear marbles total number of marbles P(clear) = = 9 20 = 0.45 = 45% The theoretical probability of drawing a clear marble is , 0.45, or 45%. 9 20

YOU TRY 1b P = P(blue) = = = 0.55 = 55% Find the probability of drawing a blue marble from the bag. number of ways the event can occur total number of equally likely outcomes P = number of blue marbles total number of marbles P(blue) = = 11 20 = 0.55 = 55% The theoretical probability of drawing a clear marble is , 0.55, or 55%. 11 20

YOU TRY 2 P = P(green) = = = 0.4 = 40% Jane has 20 marbles in a bag. Of these 8 are green. Find the probability of drawing a green marble from the bag? number of ways the event can occur total number of equally likely outcomes P = number of green marbles total number of marbles P(green) = = 8 20 = 0.4 = 40% The theoretical probability of drawing a green marble is , 0.4, or 40%. 8 20

YOU TRY 3 P = P(number more than 4) = = =  Find the probability of rolling a number more than 4 on a fair number cube. For a fair number cube, each of the six possible outcomes is equally likely. There are 2 ways to roll a number greater than 4: 5 or 6. P = number of ways the event can occur total number of equally likely outcomes P(number more than 4) = 2 numbers more than 4 6 possible outcomes = 2 6 1 3 =  0.33  33% The theoretical probability of rolling a number more than 4 is 0.33, or 33%. 1 3 ,

number of girls in the class number of students in the class YOU TRY 4 A teacher has written the name of each student on a piece of paper and placed the names in a box. She randomly draws a paper from the box to determine which student will present the answer to the problem of the day. If there are 15 boys and 12 girls in the class, what is the theoretical probability that a girl’s name will be drawn? P(girl) = number of girls in the class number of students in the class 12 27 =

What is the theoretical probability that a boy’s name will be drawn? YOU TRY 4 continued What is the theoretical probability that a boy’s name will be drawn? 15 27 P(boy) =

Find the theoretical probability of drawing a boy’s name. Extra Example There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. Find the theoretical probability of drawing a boy’s name. P(boy) = number of boys on the team number of members on the team 13 23 P(boy)= Substitute.

Extra Example continued There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. Find the theoretical probability of drawing a girl’s name. 10 23 P(girl) =

2. Find the probability of drawing a vowel. Lesson Quiz Find the probabilities. Write your answer as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. You have 11 cards, each with one of the letters from the word mathematics. 1. Find the probability of drawing an m from the pile of shuffled cards. 2. Find the probability of drawing a vowel. 3. Find the probability of drawing a consonant. 2 11 , 0.18, 18% 4 11 , 0.36, 36% 7 11 , 0.64, 64%

Lesson Quiz for Student Response Systems 1. A number cube is rolled. Identify the probability of getting a number less than 4 as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. 17 17 17

Lesson Quiz for Student Response Systems 2. There are 40 balls in a bag. Of these 8 are white, 7 are blue, 12 are green, and 13 are yellow. Identify the probability of drawing a white ball as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. 18 18 18