1. a. Miki tosses a coin 50 times, and the coin shows heads 28 times. What fraction of the 50 tosses is heads? What percent is this?
, or 56%
b. Suppose the coin is fair, and Miki tosses it 500 times. About how many times can she expect it to show heads? Explain your reasoning.
It should come up heads about 250 times, or half of 500. It will most likely not be exactly 250 heads in 500 tosses, but it is unlikely to be far from 250 heads 2. Suppose Kalvin tosses a coin to determine his breakfast cereal every day. He starts on his twelfth birthday and continues until his eighteenth birthday. About how many times would you expect him to eat Cocoa Blast cereal?
He tosses each day for about 6 years.6 × 365 days = 2,190 days (2,191 days some students might add a day or two for leap years) for which Kalvin will toss a coin. You would expect him to toss heads and eat Cocoa Blast about half of those days, or about 1,095 days
3. Kalvin tosses a coin five days in a row and gets tails every time. Do you think there is something wrong with the coin? How can you find out?
With only five trials, you cannot be certain. Kalvin should toss the coin many more times if he wants to find out whether or not the coin is fair. In fact, the probability of five consecutive heads is@ 3%. 4. Len tosses a coin three times. The coin shows heads every time. What are the chances the coin shows tails on the next toss? Explain.
The chances are , or 50%. If a coin turns up heads three times in a row, it is not more likely to turn up tails the next time, nor is it more likely to be heads again. This can be confusing for students because they expect the average to be about 50% in the short run. Experimental results are about results in the long run. 5. Is it possible to toss a coin 20 times and have it land heads-up 20 times? Is this likely to happen?
It is possible, but unlikely. Each time a coin is tossed it can land heads up, so 20 heads in a row is possible. However, there are many more possible combinations of 20 coin tosses that are not all heads, so 20 heads is very unlikely. The chance of getting 20 heads in a row is about 0.000001, that is, about 1 chance in a million.
19. Colby rolls a number cube 50 times. She records the result of each roll and organizes her data in the table below.
Number Cube Results
What fraction of the rolls are 2’s? What percent is this?
What fraction of the rolls are odd numbers? What percent is this?
What percent of the rolls is greater than 3?
Suppose Colby rolls the number cube 100 times. About how many times can she expect to roll a 2? Explain.
Answers will vary. Possible answers: 16 (the result of 100 ÷ 6); students may double the results from the given table and answer “18 times.”
If Colby rolls the number cube 1,000 times, about how many times can she expect to roll an odd number? Explain.
She can expect about 500 odd numbers, since odd and even are equally likely. Students may scale the results from this table and respond “460 times.”
6. Kalvin tosses paper cup once each day for a year to determine his breakfast cereal. Use your results from problem 1.2 to answer the following.
How many times do you expect the cup to land on its side? On one of its ends?
Answers will vary based on the experiments conducted in class. A group of students that found 20 ends in the 50 tosses should argue for a number about 7 times as large (146 ends and 219 sides in a year, 50 × 7 = 350 < 365).
How many times do you expect Kalvin to eat Cocoa Blast in a month? In a year? Explain.
Answers will vary based on the experiments conducted in class. In the example from part (a), 40% of the tosses resulted in ends, so this should happen about 12 times a month (40% of 30), or 146 times a year (40% of 365).
7. Dawn tosses a pawn from her chess set five times. It lands on its base four times and on its side only once.
Andre tosses the same pawn 100 times. It lands on its base 28 times and on its side 72 times. Based on their data, if you toss the pawn one more time, is it more likely to land on its base or its side? Explain.
The pawn is more likely to land on its side, because it is better to base a prediction on 100 tosses than on 5 tosses. It gives you even more information if you combine the data.
8. Kalvin flips a small paper cup 50 times and a large paper cup 30 times. The table below displays the results of his experiments. Based on these data, should he use the small cup or large cup to determine his breakfast each morning? Explain.
Kalvin should use the small cup and eat Cocoa Blast when it lands on its side. This is because the large cup landed on its side about 73% of the time in his experiments while the small cup landed on its side 78%
of the time. Some students may argue that the number of trials was not sufficiently large with the large cup and so the probabilities may be even closer than they appear.
For Exercises 21-23, use the bar graph below
21. Multiple Choice Suppose 41,642 people moved. About how many of those people moved for family-related reasons?
B. 11,000 22. Multiple Choice What fraction of the people represented in the graph moved for reasons other than work-related, housing-related, or family-related?
23. Multiple Choice Suppose 41,642 people moved. About how many moved for housing-related reasons?
9. Kalvin’s sister Kate finds yet another way for him to pick his breakfast. She places one blue marble and one red marble in each of two bags. She says that each morning he can choose one marble from each bag. If the marbles are the same color, he eats Cocoa Blast. If not, he eats Health Nut Flakes. Explain how selecting one marble from each of the two bags and tossing two coins are similar.
Red and blue are like heads and tails. Each bag is like a coin. Red and blue are equally likely in each bag, just as heads and tails are on each coin.
10. Adsila and Adahy have to decide who will take out the garbage. Adahy suggests they toss two coins. He says that if at least one head comes up, Adsila takes out the garbage. If no heads come up, Adahy takes out the garbage. Should Adsila agree to Adahy’s proposal? Explain why or why not.
Adsila should not agree. The probability of getting at least one head is 75%. Students can determine this by considering the possibilities, or by referring to their data from Problem 1.3 24. Suppose you write all the factors of 42 on pieces of paper and put them in a bag. You shake the bag. Then, you choose one piece of paper from the bag. Find the experimental probability of choosing the following.
a. an even number
Answers will vary. Students should describe putting all the factors, 1, 2, 3, 6, 7, 14, 21, and 42 on pieces of paper, then repeating several trials to make the experiment. Sample for 20 trials: 1, 3, 21, 42, 2, 7, 6, 2, 42, 21, 3, 3, 6,
21, 7, 7, 14, 42, 1, 42, and 21. This trial leads to a probability of even factors. Note: The theoretical probability is
Answers will vary. Using the sample data
from part (a), the probability for prime
factors is . Note: The theoretical
25. Weather forecasters often use percents to give probabilities in their forecasts. For example, a forecaster might say that there is a 50% chance of rain tomorrow. For the forecasts below, change the fractional probabilities to percents.
a. The probability that it will rain tomorrow is
40%, since , or since 2 5 = 0.40
b. The probability that it will snow Monday is
30%, since , or since 3 10 = 0.30
c. The probability that it will be cloudy this weekend is
60%, since, or since 3 5 = 0.60
For Exercises 11-15, decide whether the possible results are equally likely. Explain
For Exercises 16-17, first list all the possible results for each action. Then decide whether the results are equally likely. 16. You choose a block from a bag containing one red block, three blue blocks, and one green block.
There are three possible results—choosing a red block, a blue block, or a green block. The results are not equally likely. There are more blue blocks in the bag, so the chances of drawing a blue block are greater than the chances of drawing a red or green block. 17. You try to steal second base during a baseball game.
There are two possible results—you succeed in stealing second base, or you are out. These results are probably not equally likely. Their chances depend on the person’s skill and experience playing baseball and stealing bases. 18. For parts (a) – (f), give an example of a result that would have a probability near the percent given.
For Exercises 26-29, use the graph below
26. Is a tornado equally likely to occur in California and in Florida? Explain your reasoning.
No, a tornado is more likely to occur somewhere in Florida. 27. Is a tornado equally likely to occur in Arkansas and in Pennsylvania?
Yes, tornados are equally likely to occur in Arkansas and Pennsylvania. 28. Is a tornado equally likely to occur in Massachusetts and in Texas?
No, a tornado is more likely to occur somewhere in Texas. 29. Based on these data, is a person living in Montana more likely to experience a tornado than a person living in Massachusetts? Explain.
No. Although the data show that more tornadoes strike Montana than Massachusetts, this does not mean that a resident of Montana is more likely to experience a tornado than a resident of Massachusetts. For example, consider that Massachusetts is much smaller than Montana, and that many fewer people live in Montana.
a. Find the theoretical probability that you will choose each color.
b. Find the sum of the probabilities in part (a).
c. What is the probability that you will not choose a red block? Explain how you found your answer.
d. What is the sum of the probability of choosing a red block and the probability of not choosing a red block?
2. A bubble-gum machine contains 25 gumballs. There are 12 green, 6 purple, 2 orange, and 5 yellow gumballs.
a. Find each theoretical probability.
b. Find the sum.
P(green) + P(Purple) + P(orange) + P(yellow) =
c. Write each of the probabilities in part (a) as a percent.
d. What is the sum of all the probabilities as a percent?
e. What do you think the sum of the probabilities for all the possible outcomes must be for any situation? Explain.
3. Bailey uses the results from an experiment to calculate the probability of each color of block being chosen from a bucket. He says P(red) = 35%, P(blue) = 45%, P(yellow) = 20%. Jarod uses theoretical probability because he knows how many of each color block is in the bucket. He says P(red) = 45%, P(blue) = 35%, and P(yellow) = 20%. On Bailey’s turn, he predicts blue. On Jarod’s turn he predicts red. Neither boy makes the right prediction.
a. Did the boys make reasonable predictions based on their own probabilities? Explain.
b. Did they do something wrong with their calculations? Explain.
14. Find numbers that make the sentence true.
4. A bag contains two white blocks, one red block, and three purple blocks. You choose one block from the bag.
a. Find each probability.
b. What is the probability of not choosing a white block? Explain how you found your answer.
c. Suppose the number of blocks of each color is doubled. What happens to the probability of choosing each color?
d. Suppose you add two more blocks of each color to the original bag. What happens to the probability of choosing each color?
e. How many blocks of which colors should you add to the original bag to make the probability of choosing a red block equal to ½?
5. A bag contains exactly three blue blocks. You choose a block at random. Find each probability.
b. P(not blue)
6. A bag contains several marbles. Some are red, some are white, and some are blue. You count the marbles and find the theoretical probability of choosing a red marble is . You also find the theoretical probability of choosing a white marble is .
a. What is the least number of marbles that can be in the bag?
b. Can the bag contain 60 marbles? If so, how many of each color does it contain?
c. If the bag contains 4 red marbles and 6 white marbles, how many blue marbles does it contain?
d. How can you find the probability of choosing a blue marble?
c. The probability of an outcome can be greater than 1.
For Exercises 18-25, estimate the probability that the given event occurs. Any probability must be between 0 and 1 (or 0% and 100%). If an event is impossible, the probability it will occur is 0, or 0%. If an event is certain to happen, the probability it will occur is 1, or 100%.
The event is
The event is
certain to happen
18. You are absent from school at least one day during the school year.
19. You have pizza for lunch one day this week.
20. It snows on July 4 this year in Mexico.
21. You get all the problems on your next math test correct.
22. The next baby born in your local hospital is a girl.
23. The sun sets tonight.
24. You take a turn in a game by tossing four coins. The result is all heads.
25. You toss a coin and get 100 tails in a row.
. Patricia and Jean design a coin-tossing game. Patricia suggests tossing three coins. Jean says they can toss one coin three times. Are the outcomes different for the two situations? Explain.
9. Pietro and Eva are playing a game in which they toss a coin three times. Eva gets a point if no two consecutive toss results match (as in H – T – H). Pietro gets a point if exactly two consecutive toss results match (as in H – H – T). If all three toss results match, no one scores a point. The first player to get 10 points wins. Is this a fair game? Explain. If it is not a fair game, change the rules to make it fair.
10. Silvia and Juanita are designing a game. A player in the game tosses two number cubes. Winning depends on whether the sum of the two numbers is odd or even. Silvia and Juanita make a tree diagram of possible outcomes.
Number Cube 1
Number Cube 2
List all the outcomes for the sums.
Design rules for two-player game that is fair.
Design rules for a two-player game that is not fair.
How is this situation similar to tossing two coins and seeing if the coins match or don’t match?
26. Karen and Mia play games with coins and number cubes. No matter which game they play, Karen loses more often than Mia. Karen is not sure if she just has bad luck or if the games are unfair. The games are described in this table. Review the game rules and complete the table.
Can Karen Win?
Karen Likely to Win?
Game Fair or Unfair?
Roll a number cube.
Karen scores a point if the roll is even.
Mia scores a point if the roll is odd.
Roll a number cube
Karen scores a point if the roll is a multiple of 4.
Mia scores a point if the roll is a multiple of 3.
Toss two coins.
Karen scores a point if the coins match.
Mia scores a point if the coins do not match.
Roll two number cubes.
Karen scores a point if the number cubes match.
Mia scores a point if the number cubes to not match.
Roll two number cubes.
Karen scores a point if the product of the two numbers is 7.
Mia scores a point if the sum of the two numbers is 7.
11. Melissa is designing a birthday card for her sister. She has a blue, a yellow, a pink, and a green sheet of paper. She also has a black, a red, and a purple marker. Suppose Melissa chooses one sheet of paper and one marker at random.
a. Make a tree diagram to find all the possible color combinations.
b. What is the probability that Melissa chooses pink paper and a red marker?
c. What is the probability that Melissa chooses blue paper? What is the probability she does not choose blue paper?
d. What is the probability that she chooses a purple marker?
12. Lunch at school consists of a sandwich, a vegetable, and a fruit. Each lunch combination is equally likely to be given to a student. The students do not know what lunch they will get. Sol’s favorite lunch is a chicken sandwich, carrots, and a banana.
School Lunch Menu
a. Make a tree diagram to determine how many different lunches are possible. List all the possible outcomes.
b. What is the probability that Sol gets his favorite lunch? Explain your reasoning.
c. What is the probability that Sol gets at least one of his favorite lunch items? Explain.
13. Suppose you spin the pointer of the spinner at the right once and roll the number cube. (The numbers on the cube are 1, 2, 3, 4, 5, and 6.)
a. Make a tree diagram of the possible outcomes of a
spin of the pointer and roll of the number cube
b. What is the probability that you get a 2 on both the
spinner and the number cube? Explain your reasoning.
d. What is the probability that you get a multiple of 2 on both the number cube and the spinner?
32. Koto’s class makes the line plot shown below. Each mark represents the first letter of the name of a student in her class
First Letters of Names
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Suppose you choose a student at random from Koto’s Class.
a. What is the probability that the student’s name begins with a J?
b. What is the probability that the student’s name begins with a letter after F and before T in the alphabet?
c. What is the probability that you choose Koto?
d. Suppose two new students Melvin and Theo, join the class. You now choose a student at random from the class. What is the probability that the student’s name begins with J?
Homework What Do You Expect Investigation 3
A C E
pplications | Connections | Extensions
2. The cooks at Kyla’s school make the spinners below to help them choose the lunch menu. They let the students take turns spinning. For parts (a) – (c), decide which spinner you would choose. Explain your reasoning.
a. Your favorite lunch is pizza.
b. Your favorite lunch is lasagna.
c. Your favorite lunch is hot dogs.
3. When you use each of the spinners below, the two possible outcomes are landing on 1 and landing on 2. Are the outcomes equally likely? If not, which outcome has a greater theoretical probability? Explain.
4. Molly designs a game for a class project. She makes the three spinners shown. She tests to see which one she likes best for her game. She spins each pointer 20 times and writes down her results, but she forgets to record which spinner gives which set of data. Match each spinner with one of the data sets. Explain your answer.