Probability Lesson.1 A referee flipped a fair coin to decide which football team would start the game with the ball. The coin was just as likely to land heads as tails. Which way do you think the coin landed? Carl rolled a number cube and it landed on. There are 6 numbers on a number cube. Do you think he will roll a on his next turn? It is impossible to know for sure whether the coin will land heads or if the number cube will land on. However, you can use probability to see how likely it is each one will occur. Probability measures how likely it is an event will occur. Outcomes are the possible results you can have from a probability experiment. If you flip a coin, it can land heads or tails. If you roll a number cube, you can roll a 1, 2, 3,, 5 or 6. These are possible outcomes. An event is a desired outcome. For example, if you want to roll a, the event would be rolling a. Probabilities are measured on a scale from 0 to 1. A probability of zero means the event will never occur. A probability of one means the event will always occur. Probability 0 1_, 0.25 1_, 0.50 2 3_, 0.75 1 or 0% or 25% or 50% or 75% or 100% Impossible Unlikely Equally likely Likely Certain It is equally likely that a flipped coin will land heads or tails. Rolling a 9 on a number cube is impossible since you can only roll a 1, 2, 3,, 5 or 6. Instead of using words to describe probability, a ratio can be written to describe theoretical probability. Theoretical probability is the ratio of favorable outcomes to the possible outcomes. The notation P( ) is used for probabilities. The probability that Carl rolls a with the number cube can be written P(). This is read, The probability of rolling a. 110 Lesson.1 ~ Probability
Example 1 Solutions a. Find P(1 or 2) when rolling a number cube. b. Find P(7) when rolling a number cube. number of favorable outcomes (2 1, 2 ) a. P(1 or 2) = = _ 2 number of possible outcomes (6 1, 2, 3,, 5, 6 ) = _ 1 3 It is somewhat likely a 1 or 2 will be rolled on a number cube. number of favorable outcomes (0 7 ) b. P(7) = = _ 0 number of possible outcomes (6 1, 2, 3,, 5, 6 ) = 0 It is impossible to roll a 7 on a number cube. You can find theoretical probability without actually rolling a number cube or flipping a coin. Theoretical probability is based on what should happen. Another type of probability is experimental probability. It is based on what does happen after an experiment. Experimental probability is the ratio of the number of times an event occurs to the total number of trials. Each trial is an individual experiment. EXPLORE! coin flip Step 1: Copy the table below. Flip a coin 10 times. a. Using tallies, record the number of times the coin lands on heads and the number of times it lands on tails. Heads Tails b. Determine your experimental probability of the coin landing on heads from 10 flips by putting numbers in the following ratio. Write the probability as a fraction and a decimal. P(heads) = number of heads total number of trials Step 2: Flip the coin 10 more times (now you have flipped the coin 20 times). a. Add tallies to your chart for each additional head and tail. b. Determine your experimental probability of the coin landing on heads with 20 flips as you did in Step 1b. Write the probability as a fraction and a decimal. Lesson.1 ~ Probability 111
EXPLORE! (Continued) Step 3: Flip the coin 10 more times (now you have flipped the coin 30 times). a. Add tallies to your chart for each additional head and tail. b. Determine your experimental probability of the coin landing on heads with 30 flips as you did in Step 1b. Write the probability as a fraction and a decimal. Step : Repeat this process until you have flipped the coin 100 times. Stop after every 10 flips to record your results and experimental probability of the coin landing on heads. Step 5: Find the theoretical probability of the coin landing on heads. How many heads would you expect to get after 100 flips? Step 6: After which set of flips was the probability closest to 0.5? Step 7: Why do you think doing more trials makes the experimental probability closer to the theoretical probability? Example 2 Solutions Kyle rolled a number cube 60 times. His results are shown in the table below. Number Rolled 1 2 3 5 6 Frequency (number of times rolled) 8 12 6 15 15 a. Find Kyle s experimental probability of rolling a 6. b. Find Kyle s experimental probability of not rolling a 6. c. Find the theoretical probability of rolling a 6. d. Find the theoretical probability of not rolling a 6. a. P(6) = number of times a 6 was rolled (15) = 15 number of trials (60) 60 = _ 1 b. P(not 6) = number of times a 6 was not rolled (8 + 12 + 6 + 15 + ) = 5 number of trials (60) 60 = _ 3 c. P(6) = number of favorable outcomes (1 6 ) number of possible outcomes (6 1, 2, 3,, 5 or 6 ) = 1 _ d. P(not 6) = number of favorable outcomes (5 1, 2, 3, or 5 ) number of possible outcomes (6 1, 2, 3,, 5 or 6 ) = _ 5 P(6) and P(not 6) are called complements of each other because together they make up all the possible outcomes without repeating any outcomes. This means P(6) and P(not 6) sum to 1. You can find P(not 6) by subtracting P(6) from 1. P(6) = 1 _ 112 Lesson.1 ~ Probability P(not 6) = 1 1 _ = 5_ Sometimes probabilities are written as decimals or percents. This is the same as rewriting a fraction as a decimal or a percent.
Example 3 Use the spinner to write each probability as a fraction, decimal and percent. a. P(yellow) b. P(green) c. P(red or green) Solutions Fraction Decimal Percent a. P(yellow) = 1_ a. 1_ = 0.25 a. 0.25 100 = 25% b. P(green) = 2_ = 1_ 2 b. 1_ = 0.5 b. 0.5 100 = 50% 2 c. P(red or green) = 3_ c. 3_ = 0.75 c. 0.75 100 = 75% exercises 1. Give an example of an event which has a probability of 0. Explain why the probability equals 0. 2. Give an example of an event which has a probability of 1. Explain why the probability equals 1. Find each probability for one roll of a number cube. Write as a fraction in simplest form. 3. P(3). P(1, or 6) 5. P(0) 6. P(even number) 7. P(not ) 8. P(2) A game has 30 cards numbered 1 through 30. The cards are shuffled. One card is picked at random. Find each probability. 9. P(20) 10. P(multiple of 5) 11. P(even number) 12. P(31) 13. P(10, 20 or 30) 1. P(number greater than 20) 15. Which cards would give a probability of 1 10 they were picked? 16. Which cards would give a probability of 1 they were picked? A letter is chosen at random from the word PROPORTION. Find each probability. 17. P(T) 18. P(R) 19. P(O) 20. P(vowel) 21. P(E) 22. P(consonant) 23. a. What is the theoretical probability of a flipped coin landing tails? b. You flip a coin 0 times. How many times would you expect it to land tails based on theoretical probability? c. Are you guaranteed to get tails 20 times? Explain your reasoning. Lesson.1 ~ Probability 113
2. Sarah surfs with her friends at a beach in Hawaii. Sarah caught waves out of 20 possible waves to ride in to shore. Find the experimental probability Sarah will catch a wave. 25. A professional basketball player was practicing free throws before a game. He made 12 of 15 shots. Find the experimental probability he will make a free throw during the game based on his practice shots. 26. A game requires each player to roll two number cubes and sum the results. Copy and complete the chart to find all the possible sums. The top row and left column are the numbers rolled on each number cube. The interior of the chart shows the corresponding sums. a. Find P(sum of 2). b. Find P(sum of 10). c. In order to win the game, a player needs to roll a sum of 7. Find P(sum of 7). d. Shelby has had 30 turns and still has not rolled a sum of 7. Do you think the number cubes are fair? Explain your reasoning. Number Cube Sum + 1 2 3 5 6 1 2 3 5 6 7 2 3 5 6 7 8 3 5 6 Use the given spinner to find each probability. Write each probability as a fraction, decimal and percent. 27. P(blue) 28. P(even number) 8 1 7 2 29. P(red) 30. P(6) 31. P(9) 6 5 3 32. Write two different probabilities from the spinner that have the same value. Show that the values of the probabilities are equal. Copy the checkerboard on your paper for each Exercise 33 36. Color it so the given probabilities are true if a dart is thrown at random and lands somewhere on the board. 33. P(yellow) = 1 12 3. P(blue) = 1 _ 2 35. P(red) = 1 _ 36. P(green) = 1 _ 37. There are 10 red jelly beans, 6 pink jelly beans and yellow jelly beans in a jar. After each person picks a jelly bean without looking, it is put back in the jar before the next person picks. a. How many jelly beans are in the jar? b. Jenny reaches in and grabs one jelly bean. Find P(red). c. Justin reaches in and grabs one jelly bean. Find P(pink). d. Nancy reaches in and grabs one jelly bean. Find P(yellow). e. Which color is most likely to be chosen? Explain your reasoning. 11 Lesson.1 ~ Probability
review Write each fraction as a decimal and a percent. 38. 1_ 39. 1_ 2 0. 3_ Write each decimal as a fraction and a percent. 1. 0.5 2. 0.8 3. 0.75. Order the numbers 0.67, 3_ 5, 2_ 3 and 66% from least to greatest. Show all work necessary to justify your answer. Tic-Tac-Toe ~ Buffon s Ne e dle Buffon s needle problem was first posed by 18th century mathematician Georges-Louis Leclerc, Comte de Buffon. It was one of the first geometric probability problems solved and has since become a simulation to approximate the value of the number π (pi 3.1159). Find a toothpick, a blank piece of paper, a ruler and a ballpoint pen. Draw straight horizontal lines across the paper using the ruler so the lines are the same distance apart as the length of the toothpick. Step 1: Drop the toothpick onto the paper and record whether or not the toothpick is crossing one of the lines on the paper. Copy the table below to record your results. Drop the toothpick a total of 100 times. Line Crossed No Line Crossed Step 2: Find the experimental probability a line is crossed when the toothpick is dropped. The theoretical probability the toothpick will cross a line is _ 2 or about 0.6366. How close is your experimental π probability to the theoretical probability? Step 3: From the theoretical probability, you can estimate the value of π by calculating: 2 total number of toothpick drops π= number of times a line is crossed Approximate π using your 100 toothpick drops. How close is your value to π 3.1159? Step : Drop the toothpick 100 more times and add the results to your first 100 drops. Find the experimental probability and a new approximation for π. Lesson.1 ~ Probability 115