Lesson Lesson 3.7 ~ Theoretical Probability


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1 Theoretical Probability Lesson.7 EXPLORE! sum of two number cubes Step : Copy and complete the chart below. It shows the possible outcomes of one number cube across the top, and a second down the left column. The corresponding sums are shown in the table. Fill in the missing sums a. How many sums are shown in the chart? b. Find and record the frequency of each sum (the number of times each sum appears). Sum Step : Copy and complete the table below to find the probability of rolling each sum on this chart. Sum Probability number of ways to get a sum of 7 For example, to find the probability of rolling a 7, find number of outcomes possible = = _. Step : Which sum is most likely to occur? Step : Roll two number cubes times. Copy the table and record the sums with tally marks. Sum Step : Find the experimental probability of rolling a sum of 7 on your next turn. Does your experimental probability match the probability in Step? Explain your reasoning. You found probabilities by doing experiments in the last lesson. Sometimes the outcomes did not have the same probability as you would have expected. When flipping a coin, you would expect to get as many heads as tails. This is called theoretical probability. Theoretical probability is the expected ratio of favorable outcomes (wanted outcomes) to the number of possible outcomes. 00 Lesson.7 ~ Theoretical Probability
2 Example Solutions a. Find P(tails) when flipping a coin. b. Find P( or ) when rolling one number cube. c. Find P(not ) when rolling one number cube. a. P(tails) = number of favorable outcomes (tails) number of possible outcomes (heads or tails) = _ b. P( or ) = number of favorable outcomes (, ) number of possible outcomes (,,,,, ) = _ = _ c. P(not ) = number of favorable outcomes (,,,, ) number of possible outcomes (,,,,, ) = _ You will always get a head or tail when you flip a coin, so P(heads or tails) =. Also, when rolling a number cube, P(,,,, or ) = since those are the only outcomes possible. Another way to find P(not ) when rolling a number cube is determining P() and subtracting the answer from. P(not ) = P() = _ = _ This is the same as the answer in Example c. P(not ) and P() are called complements because together they contain all possible outcomes with no overlap. The sum of complements is always. Example Each letter from the word MATHEMATICS is written on a separate card. A card is chosen at random. Find the probability of each event. a. P(C) b. P(M) c. P(not M) d. P(vowel) Solutions a. P(C) = C letters possible = b. P(M) = Ms letters possible = c. P(not M) = P(M) = = 9 A, E, A, I d. P(vowel) = letters possible = Lesson.7 ~ Theoretical Probability 0
3 Exercises Find each probability for one roll of a number cube. Write each answer as a fraction in simplest form.. P(). P(, or ). P(even number). P(0). P(not ). P(less than 7) Each letter from the word PROBABILITY is written on a separate card. A card is chosen at random. Find the probability of each event. Write each answer as a fraction in simplest form. 7. P(P) 8. P(not P) 9. P(vowel) 0. P(consonant). P(B or I). P(Q) You play a game with 0 cards numbered through 0. The cards are shuffled and one is picked at random from the complete deck. Find the probability of each event as a fraction in simplest form.. P(). P(0). P(odd number) Use the spinner to find each probability. Write each probability as a fraction, decimal and percent.. P(blue) 7. P(even number) P(red) 9. P(not green) 0. P(9). Write two different probabilities from the spinner that have the same value. What is the value?. Jermaine created his own card game. The probability of getting a blue card was _. Find P(not blue).. Bena has seven cats. Three of the cats are black. One of the cats was chosen at random. What is the probability that it is not black?. The probability of landing on the purple region of a spinner is. Find P(not purple). 0 Lesson.7 ~ Theoretical Probability. Todd had a bag of marbles containing blue, green and yellow marbles. He reached into the bag and pulled out a yellow marble. He did not return the yellow marble to the bag. a. Find the probability the second marble drawn will also be yellow. b. Find the probability the second marble drawn will be green.
4 . A game required each player to roll two number cubes and find their sum. Complete the chart below to find all possible sums. The top row and left column are the possible numbers rolled on each number cube. The inside of the chart shows the corresponding sum. Find: Number Cube Sums a. P() b. P(not ) + 7 c. P( or ) 7 8 d. P(even) e. P(0) f. P(less than ) g. Carrie rolls the number cubes times trying to get a sum of 7 to win the game. She never gets a sum of 7. Do you think the number cubes are fair? Explain your reasoning. 7. Copy the spinner on your paper. Color it so each probability is true when a person spins it. a. P(green) = _ b. P(blue) = _ c. P(red) = d. P(yellow) = _ review 8. Addison rolled a number cube 0 times. It landed on four times. a. Find the experimental probability she will roll a on her next roll. b. Find the theoretical probability she will roll a on her next roll. 9. Sai had a bag of marbles: blue, 8 red, yellow and green. He chose one marble from the bag. a. Find the theoretical probability Sai will choose a blue marble. b. After choosing and returning marbles 0 times, Sai chose a blue marble times. Find the experimental probability Sai will choose a blue marble on his next pick. 0. There is a 0% chance of rain. Which phrase best describes the chance of rain? impossible, unlikely, equally likely, likely, certain Solve each percent problem.. 0% of 800 is. % of 00 is. % of 0 is Lesson.7 ~ Theoretical Probability 0
5 TicTacToe ~ Numbe r Cube Difference s Step : Copy and complete the chart below which shows one number cube across the top, the second down the left column and the corresponding differences in the middle. Fill in the missing differences. Always subtract the smaller number from the larger number a. How many differences are shown in the chart? b. Find the frequency of each difference (the number of times each difference appears). Difference 0 Step : Copy and complete the chart below. Find the theoretical probability of rolling each difference on this chart. Example: Find the theoretical probability of rolling a difference of. number of ways to get a difference of = 0 total number of outcomes possible = 8 Difference 0 Probability as a fraction Probability as a decimal Probability as a percent 8 Step : Which difference is most likely to occur? Step : Roll two number cubes times. Copy the table and record the differences with tally marks. Difference 0 Experimental probability Step : a. Find the experimental probability of rolling each difference. b. How does your experimental probability compare to the theoretical probability in Step? Explain why there may be similarities or differences. 0 Lesson.7 ~ Theoretical Probability
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