3.6: Exercises
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A random experiment consists of drawing a single card from a well-shuffled deck and recording the suit. Write the sample space for this experiment.
A random experiment consists of drawing a single card from a well-shuffled deck and recording the number. Write the sample space for this experiment.
A random experiment consists of tossing a fair coin five times and recording the number of tails. Write the sample space for this experiment.
A random experiment consists of tossing a fair coin four times and recording the number of heads. Write the sample space for this experiment.
A random experiment consists of tossing three fair coins and recording whether each coin lands heads up or tails up. Write the sample space for this experiment.
A random experiment consists of tossing four fair coins and recording whether each coin lands heads up or tails up. Write the sample space for this experiment.
A random experiment consists of tossing a fair die and then tossing a fair coin. The number showing on top of the die and whether the coin lands heads up or tails up is recorded. Write the sample space for this experiment.
A spinner in the following figure is spun and a coin is tossed. The color of the spinner and whether the coin is heads up or tails up is recorded. Write the sample space for this experiment.
A single card is drawn from a well-shuffled deck of 52 cards. Find the probability that:
- the card is a seven.
- the card is a face card.
- the card is a number between 2 and 5, inclusive.
A single card is drawn from a well-shuffled deck of 52 cards. Find the probability that:
- the card is a jack.
- the card is a club.
- the card is a number less than 4, including aces.
Two fair dice are tossed. What is the probability that the sum of the numbers is 5?
Two fair dice are tossed. What is the probability that the sum of the numbers is 10?
The spinner is spun once. What is the probability that it lands on green?
The spinner is spun once. What is the probability that it lands on blue?
An urn contains 10 red balls, 15 white balls, and 20 black balls. A single ball is selected at random. What is the probability that the ball is:
- red?
- white?
- not black?
- black or white?
A candy dish contains 12 chocolate candies, 18 butterscotch candies, eight caramels, and 15 peppermints. A single candy is selected at random. What is the probability that the candy is:
- a butterscotch candy?
- not a caramel candy?
- a chocolate or a peppermint candy?
Three fair coins are tossed at the same time. What is the probability of getting exactly two heads? Hint: use problem #5.
Four fair coins are tossed at the same time. What is the probability of getting exactly one head? Hint: use problem #6.
An instructor collected the following data from the students in her classes.
| Class / Year | Freshman | Sophomore | Total |
| MAT 121 | 43 | 15 | 58 |
| MAT 142 | 35 | 28 | 63 |
| MAT 187 | 27 | 32 | 59 |
| Total | 105 | 75 | 180 |
If a student is selected at random, what is the probability that:
- The student is a freshman?
- The student is taking MAT 142?
A real estate agent has kept records of the number of bedrooms in the houses he has sold for the last five years. The data is listed in the following table.
| Year/ Number of Bedrooms | one or two | three | four | five or more | Total |
| 2009 | 5 | 12 | 25 | 3 | 45 |
| 2010 | 7 | 15 | 22 | 3 | 47 |
| 2011 | 6 | 18 | 28 | 6 | 58 |
| 2012 | 6 | 16 | 30 | 2 | 54 |
| 2013 | 5 | 17 | 28 | 4 | 54 |
| Total | 29 | 78 | 133 | 18 | 258 |
- Find the probability that a randomly selected house sold by the agent had four bedrooms.
- Find the probability that a randomly selected house sold by the agent was sold in 2010.
- Find the probability that a randomly selected house sold by the agent had less than four bedrooms.
- Find the probability that a randomly selected house sold by the agent was sold after 2011.
A student believes he has an 80% chance of passing his English class. What is the probability he will not pass the class?
There is a 45% chance that it will snow today. What is the probability that it will not snow today?
A single die is rolled. What is the probability of not rolling a five?
A student is randomly chosen from a class of 30 students. If seven of the students are majoring in business, what is the probability that the randomly chosen student is not majoring in business?
There is a 35% chance that a bus will arrive early at a bus stop. Explain why you cannot assume that there is a 65% probability that the bus will be late at the same bus stop.
Are each of the following valid probability distributions or not? For each one, explain why or why not.
1.
| Outcome | A | B | C | D | E |
| Probability | 0.2 | 0.4 | -0.2 | 0.4 | 0.2 |
2.
| Outcome | A | B | C | D | E |
| Probability | 0.2 | 0.4 | 0.1 | 0.4 | 0.2 |
3.
| Outcome | A | B | C | D | E |
| Probability | 0.20 | 0.30 | 0.10 | 0.15 | 0.25 |
Three fair coins are tossed and the number of heads recorded. Write the probability distribution for this random experiment. Hint: use problem #5.
Four fair coins are tossed and the number of heads recorded. Write the probability distribution for this experiment. Hint: use problem #6.
The spinner in the following figure is spun once and the color recorded. Write the probability distribution for this random experiment.
An urn contains three red balls, four blue balls, and five green balls. A ball is selected at random and its color recorded. Write out the probability distribution for this experiment.
A single card is drawn from a well-shuffled deck of 52 cards. The card is either an even number, an odd number or a face card. If we consider aces to be ones, write the probability distribution for this random experiment.
Two fair dice are rolled. Find the odds for rolling a sum of five.
Two fair dice are rolled. Find the odds for rolling a sum of 10.
Two fair dice are rolled. Find the odds against rolling a sum of six.
Two fair dice are rolled. Find the odds against rolling a sum of eight.
A single card is drawn from a well-shuffled deck of 52 cards. Find the odds that the card is a nine.
A single card is drawn from a well-shuffled deck of 52 cards. Find the odds against the card being a face card.
A candy dish contains 12 chocolate candies, 18 butterscotch candies, 8 caramels, and 15 peppermints. A single candy is selected at random. What are the odds that the candy is a chocolate candy?
An urn contains 10 red balls, 15 white balls, and 20 black balls. A single ball is selected at random. Find the odds against drawing a white ball.
A real estate agent has kept records of the number of bedrooms in the houses he has sold for the last year. The data is listed in the following table.
| Number of Bedrooms | 1 or 2 | 3 | 4 | 5 or more |
| Number of Houses | 5 | 12 | 25 | 3 |
If a house sold by the agent is selected at random, find:
- the odds for the house having four bedrooms.
- the odds for the house having less than three bedrooms.
- the odds against the house having five or more bedrooms.
- the odds against the house having more than three bedrooms.
Suppose the odds that Josh will win a tennis match against Jesse are 2 to 1. What is the probability that Josh will win?
The odds of winning a particular carnival game are 27 to 5. Find the probability of winning the game.
The odds of winning a particular carnival game are 15 to 28. Find the probability of losing the game.
The odds that a randomly selected student is a male are 21 to 25. Find the probability that a randomly selected student is a male.
A single card is drawn from a well-shuffled deck of 52 cards. Are the events E = the card is an even number and F = the card is a heart mutually exclusive? Explain why or why not.
A single card is drawn from a well-shuffled deck of 52 cards. Are the events E = the card is a face card and F = the card is a seven mutually exclusive? Explain why or why not.
An instructor randomly selects a student from his class. Are the events E = the student is taking History and F = the student is a freshman mutually exclusive? Explain why or why not?
A campus security officer randomly selects a car in the parking lot. Are the event E = the car is a Toyota and F = the car is red mutually exclusive? Explain why or why not.
Suppose you roll a single fair die. What is the probability of getting a four or a five?
Suppose you roll a single fair die. What is the probability of getting a two or an even?
Suppose you draw one card from a standard deck of cards. What is the probability that the card is an ace or a diamond?
Suppose you draw one card from a standard deck of cards. What is the probability that the card is an ace or the king of diamonds?
A teacher asks a class of 40 students about the classes they are taking. 17 of the students are taking math, 31 are taking English and 15 are taking both math and English. What is the probability that a randomly selected student is taking either math or English?
A teachers looks over her class and notices a few trends. Out of the 60 students in the class, 23 have brown hair, seven have green eyes, and three have both brown hair and green eyes. What is the probability that a randomly selected student has either brown hair or green eyes?
A student observes 46 vehicles in the CCC parking lot. He notices that 12 of the vehicles are red and that 19 of the vehicles are 4-wheel drive. If the probability that a randomly chosen vehicle is red or has 4-wheel drive is 0.609, what is the probability that the car is red and has 4-wheel drive?
Are drawing a card from a deck and tossing a coin independent events? Explain why or why not.
A single card is drawn from a deck. Are drawing a red card and drawing a heart independent events? Explain why or why not.
A jar contains three red, four blue, and five white marbles. A marble is drawn and its color is recorded. The marble is put back in the jar and a second marble is drawn. Is drawing two marbles in this manner independent events or not? Explain why or why not.
A jar contains three red, four blue, and five white marbles. A marble is drawn and its color is recorded. The marble is not put back in the jar before a second marble is drawn. Is drawing two marbles in this manner independent events or not? Explain why or why not.
A fair coin is tossed ten times. What is the probability of getting ten heads?
A fair coin is tossed then a fair die is rolled. What is the probability of getting a head and a number less than three?
A card is drawn from a well-shuffled deck, its number recorded, and the card returned to the deck. A second card is then drawn.
- What is the probability of getting a jack first and a spade second?
- What is the probability of getting a jack and a spade in any order?
The spinner shown in the following figure is spun three times. What is the probability that the spinner lands on red first, blue second, and yellow third?
The spinner shown in the following figure is spun four times. What is the probability that the spinner lands on red the first three times and blue the fourth time?
A mother figures that her son will forget his homework about 20% of the time. What is the probability that he will forget his homework at least once in the next 10 school days?
A bird watcher expects to see a falcon about 35% of the times he visits a nearby park. If he visits the park 12 times in the next month, what is the probability that he will see a falcon at least once?
A police officer watching a particular stretch of highway in Montana finds that about one out of every five drivers is speeding. What is the probability that at least one of the next eight cars he sees is speeding?
An instructor expects 10% of the class to earn an A on the final exam. If there are 23 students in the class, what is the probability that at least one student earns an A on the final exam?
A researcher surveys 150 young athletes asking about the last sport the athlete played and whether or not the athlete was injured playing that sport. The data is summarized in the following table.
| Sport | Injured | Not Injured | Total |
| Gymnastics | 16 | 34 | 50 |
| Soccer | 5 | 30 | 35 |
| Football | 27 | 18 | 45 |
| Skiing | 7 | 13 | 20 |
| Total | 55 | 95 | 150 |
If an athlete is selected at random, what is the probability that the athlete:
- played soccer and was not injured?
- did gymnastics and was injured?
- played football or was injured?
An instructor collected the following data from the students in her classes.
| Class / Year | Freshman | Sophomore | Total |
| MAT 121 | 43 | 15 | 58 |
| MAT 142 | 35 | 28 | 63 |
| MAT 187 | 27 | 32 | 59 |
| Total | 105 | 75 | 180 |
If a student is selected at random, what is the probability that:
- the student is a freshman and taking MAT 121?
- the student is a sophomore and taking MAT 187?
- the student is a freshman or taking MAT 142?
Suppose you draw two cards from a standard deck of cards without replacement. What is the probability that:
- both cards are Kings?
- both cards are face cards?
- the first card is a five and the second card is a Jack?
- the first card is a Queen and the second card is a number less than five (count Aces as ones)?
Suppose you draw two cards from a standard deck of cards without replacement. What is the probability that:
- you draw an Ace on the first card and a seven on the second card?
- you draw a heart on the first card and a spade on the second card?
- you draw an eight on the first card and a face card on the second card?
- you draw two hearts in a row?
Suppose you pick two candies randomly from a box of candies and eat them. There are four chocolates, four caramels, and four mints. What is the probability that both candies will be chocolates?
A kindergarten class has 15 boys and 13 girls. The teacher calls on three students, one at a time, to line up at the board. What is the probability that the teacher calls up a boy followed by two girls?
A small parking lot has five black cars, seven white cars, three red cars, and four blue cars. If two cars leave in random order, what is the probability that a red car will leave first, followed by a black car?
A cooler contains six colas, eight root beers, and four ginger ales. Three kids grab a drink at random, one at a time.
- What is the probability that the first kid grabs a cola, the second kid grabs a ginger ale, and the third kid grabs a cola?
- What is the probability that the third kid grabs a root beer given that the first two grabbed colas?
An instructor collected the following data from the students in her classes.
| Class / Year | Freshman | Sophomore | Total |
| MAT 121 | 43 | 15 | 58 |
| MAT 142 | 35 | 28 | 63 |
| MAT 187 | 27 | 32 | 59 |
| Total | 105 | 75 | 180 |
If a student is selected at random, find:
- the probability that the student is a freshman given the student is taking MAT 121.
- the probability that the student is taking MAT 142 given that the student is a sophomore.
- the probability that the student is a freshman and taking MAT 187.
A real estate agent has kept records of the number of bedrooms in the houses he has sold for the last five years. The data is listed in the table.
| Year/ Number of Bedrooms | one or two | three | four | five or more | Total |
| 2009 | 5 | 12 | 25 | 3 | 45 |
| 2010 | 7 | 15 | 22 | 3 | 47 |
| 2011 | 6 | 18 | 28 | 6 | 58 |
| 2012 | 6 | 16 | 30 | 2 | 54 |
| 2013 | 5 | 17 | 28 | 4 | 54 |
| Total | 29 | 78 | 133 | 18 | 258 |
If a house sold by the agent is randomly selected, find:
- the probability that the house has three bedrooms given it was sold in 2012.
- the probability that the house has five or more bedrooms and was sold in 2009.
- the probability that the house has four bedrooms and was sold in 2013.
- the probability that the house was sold in 2011 given that is has four bedrooms.
A single card is drawn from a well-shuffled deck of 52 cards. If the card is an ace, you win $12; otherwise, you lose $1.50.
- What is the expected value of this game?
- Explain what the expected value means in terms of the game.
- Is this a fair game or not?
Four thousand tickets are sold at $1 each for a charity raffle. Tickets are drawn at random without replacement. There is one $800 first prize, two $300 second prizes and eight $50 third prizes.
- What is the expected value of this raffle if you buy one ticket?
- Explain what the expected value means in terms of the raffle.
- Is this a fair game or not?
You and a friend are playing some of the games at the county fair. A particular game consists of drawing a single card from a standard deck. You win $5 if you draw an ace, $2 if you draw a face card, and $0.25 if any other card is drawn. It costs $1 to draw a card. Your friend thinks it is a great idea to play this game.
- Calculate the expected value for this game.
- Should your friend play the game or not?
- Using simple English explain to your friend why he should or should not play the game.
A casino game has an expected value of -$0.15. Explain what this means in a complete sentence. Do not use the words “expected value” in your explanation.
Two coins are flipped. You win $3 if either two heads or two tails turns up; you lose $4 if a head and a tail turn up. What is the expected value of the game?
Two dice are tossed at the same time. If the sum of the numbers is less than eight you win $5.50; if the sum is exactly eight you win $8; and if the sum is greater than eight you win $3. It costs $5.00 to play the game. Use the expected value to determine if you should play this game or not. Explain why or why not.
A lottery game consists of picking five numbers between 1 and 36, inclusive. You want to buy a ticket with the numbers 1, 2, 3, 4, and 5. Your friend laughs at you and says that those five numbers are really unlikely. What should you say to your friend to explain why he is wrong?
You and your friend are playing some slot machines in a casino. Your friend has lost the last 15 times he played. He asks to borrow some money because he feels his luck is about to change, that he is due to win after all those loses. What should you say to your friend to explain why he is wrong?
Suppose you want to create a computer password that has to be 10 digits long with each digit being chosen from the numbers 0 through 9. How many different passwords are available?
Jack wants to buy a new set of yard furniture. If there are three choices of tables, five choices of gliders, six choices of cushions, and two choices of umbrellas, how many different sets of yard furniture are possible?
The chess club is having a Sundae Night to raise money. There are three flavors of ice cream and six different toppings. A sundae consists of two scoops of ice cream and one topping. The ice cream scoops do not have to be the same flavor. How many different sundaes are possible?
The new iPhone comes in five different colors. There are three different sizes of memory available. The store has a selection of ten different cases to fit it. Assuming a person wants to buy an iPhone and a case, how many different phone/case combinations are possible?
How many ways can you line up 5 people?
Ten runners participate in the 100 meter hurdles. In how many ways can the medals for the top three finishers be awarded?
The Science club has 25 members. In how many ways can a President, Vice President, Secretary, and Treasurer be chosen?
Five friends are taking a road trip in a car that seats five. The car has a standard transmission so only two of the friends can drive it. How many seating arrangements are possible?
A night club owner is trying to arrange nine acts for a show. There are five musical numbers and four comedians. In how many ways can the show be arranged if:
- the acts can be in any order?
- the acts must alternate between musical numbers and comedians, starting with a musical number?
- the show must open and close with musical numbers but the remaining acts may be in any order?
Stanley wants to rearrange six science books and four history books on his shelf. How many arrangements are possible if:
- the books can be in any order?
- the science books are on the left and the history books are on the right?
- the four history books are in the middle with three science books on each end?
In how many ways can a five-card poker hand be drawn from a standard deck of cards?
In how many way can a two-card poker hand be drawn from a standard deck of cards?
The science club consists of 18 men and 12 women. Five members are chosen to staff the club’s booth at the Science in the Park Festival. In how many ways can the five members be chosen if:
- any of the members can be chosen?
- all five members are women?
- exactly three men are chosen?
There are 28 graduate students in the department of Math and Statistics. Eighteen of the students are majoring in math and the other 10 are majoring in statistics. The department wants to send four of the graduate students to a conference. How many ways can the four students be chosen if:
- any of the students can be chosen?
- two students from each major must be chosen?
- at least two must be majoring in math.
A barrel contains 20 good peaches and four rotten peaches. A person selects three peaches at random. In how many ways can the person select:
- three rotten peaches?
- three good peaches?
- two good and one rotten peach?
A toddler is playing with some magnetic letters on the refrigerator. He has the letters l, t, a, b, and e. If he arranges the letters in a line to make a word, what is the probability that he makes the word “table”?
The theatre club has 14 female and nine male members. Two members are selected at random to do an acting exercise. What is the probability that both members are males?
Three friends, Al, Ted, and Bert run a foot race with five other boys. What is the probability that:
- Ted finishes first, Bert finishes second and Al finishes third?
- the three friends all finish in the top three places?
A night club owner is trying to arrange nine acts for a show. There are five musical numbers and four comedians. If the owner randomly arranges the acts, what is the probability that:
- the acts alternate between musical numbers and comedians, starting with a musical number?
- the show opens and closes with musical numbers?
The science club consists of 18 men and 12 women. Five members are chosen to staff the club’s booth at the Science in the Park Festival. What is the probability that:
- all five members are women?
- exactly three men are chosen?
There are 28 graduate students in the department of Math and Statistics. Eighteen of the students are majoring in math and the other ten are majoring in statistics. The department wants to send four of the graduate students to a conference. What is the probability that:
- two students from each major are chosen to attend the conference?
- at least two who are majoring in math are chosen to attend the conference?
A barrel contains 20 good peaches and four bad peaches. A person selects three peaches at random. what is the probability of getting:
- three bad peaches?
- three good peaches?
- two good and one bad peach?