Unit 7
An insurance policy sells for $1200. Based on past data, an average of 1 in 50 policyholders will file a $20,000 claim, an average of 1 in 200 policyholders will file a $50,000 claim, and an average of 1 in 250 policyholders will file an $80,000 claim. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss?
$230 profit, $2,300,000
Use the theoretical method to determine the probability of the following event. Sharing a birthday with another person when you both have birthdays in June.
1/30
The probability of drawing either a non-face card or a queen from a regular deck of cards is
11/13
The next six births at a hospital all being boys.
The individual events are independent. The probability of the combined event is 1/64
Determine the probability of having 1 girl and 3 boys in a 4-child family assuming boys and girls are equally likely.
The probability of having 1 girl and 3 boys is 1/4.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I estimate that the probability of my getting married in the next 3 years is 0.7.
The statement makes sense. The statement is an example of subjective probability.
According to the video in a new tab, how much should one expect to lose if you purchase 1000 lottery tickets?
$4350
The odds on (against) your bet are 1 to 4. If you bet $36 and win, how much will you gain?
$9
Suppose that you arrive at a bus stop randomly, so all arrival times are equally likely. The bus arrives regularly every 60 minutes without delay. What is the expected value of your waiting time? Explain.
-30 minutes -mean
An insurance policy sells for $1200. Based on past data, an average of 1 in 100 policyholders will file a $15,000 claim, an average of 1 in 250 policyholders will file a $30,000 claim, and an average of 1 in 400 policyholders will file a $70,000 claim. Find the expected value (to the company) per policy sold. If the company sells 20,000 policies, what is the expected profit or loss?
-755 -profit 15100000 (expected value multiplied by number of policies.
An insurance policy sells for $1200. Based on past data, an average of 1 in 100 policyholders will file a $10,000 claim, an average of 1 in 200 policyholders will file a $30,000 claim, and an average of 1 in 400 policyholders will file a $70,000 claim. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or loss?
-775 -profit 7,750,000
According to the video in a new tab, what is the probability of spinning either A or B?
1/2
The odds on (against) your bet are 2 to 7. If you bet $63 and win, how much will you gain?
18 ($63 divided by 2/7 = 18)
Drawing either an ace or a ten from a standard deck of cards
2/13
According to the video in a new tab, what is the theoretical probability of getting a green number on any spin?
2/38
Use the theoretical method to determine the probability of the outcome or event given below. The next president of the United States was born on Sunday or Saturday.
2/7
The local weather forecast has been accurate for 20 of the past 37 days. Based on this fact, what is the relative frequency probability that the forecast for tomorrow will be accurate?
20/37=0.541
Use the "at least once" rule to find the probabilities of the following event. Getting at least one head when tossing five fair coins
31/32
Determine the probability of the given opposite event. What is the probability of rolling a fair die and not getting an outcome less than 2?
5/6
Use the "at least once" rule to find the probabilities of the following event. Getting at least one head when tossing nine fair coins
511/512
What does it mean when we write P(A)? What is the possible range of values for P(A), and why?
P(A) means the probability that event A will occur.
Briefly explain why quantifying risk is important to decision making. Why is it important to quantify risk when making a decision?
Quantifying risk is important so it can be measured against the benefits when making decisions appropriate for our own personal circumstances.
How did the gambling habits of Chevalier de Mère help launch the mathematical study of probability?
The Chevalier calculated his chance of winning incorrectly, but still won a lot of money. When he began losing money, he turned to a mathematician to figure out his error.
An insurance company knows that the average cost to build a home in a new California subdivision is $97,003 and that in any particular year there is a 1 in 51 chance of a wildfire destroying all the homes in the subdivision. Based on these data and assuming the insurance company wants a positive expected value when it sells policies, what is the minimum the company must charge for fire insurance policies in this subdivision?
The company must charge $1902 per year because this is the expected value.
The table shows the leading causes of death in a certain country in a recent year. The population of the country was 315 million. What is the empirical probability of death by heart disease during a single year? How much greater is the risk of death by heart disease than death by suicide?
The empirical probability of death by heart disease during a single year is .00189 (Divide population by Number) -The risk of death by heart disease is about enter your response here times greater than risk by death of suicide. 15.8
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Randomly selecting a four-person committee consisting entirely of women from a pool of 16 men and 14 women.
The event of selecting a woman and the event of selecting a woman the next time are dependent. -The probability of randomly selecting a four-person committee consisting entirely of women from a pool of 16 men and 14 women is
Rolling two 4s followed by one 1 on three tosses of a fair die.
The individual events are independent. The probability of the combined event is 1/6x1/6x1/6=1/216.
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. The next eight births at a hospital all being girls.
The individual events are independent. The probability of the combined events is 1/256
Drawing either a non-face card or a queen from a regular deck of cards.
The individual events are non-overlapping. 11/13
You roll two dice twice. Based on the probabilities shown in the table, what is the probability that you'll get a sum of 3 on the first roll and a sum of 4 on the second roll? Event (Sum) 2 3 4 5 6 7 8 9 10 11 12 Total Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 1
The probability is 2/36×3/36 because the two events are independent.
Each of the following states a probability for two events. In which case are the events dependent?
The probability of choosing two pink roses from a vase with 25 roses. After the first rose is picked, there will be one less pink rose and one less out of the total in the vase.
Use the method of your choice to determine the following probability. Drawing three queens in a row from a standard deck of cards when the drawn card is not returned to the deck each time.
The probability of drawing three queens is 1/5525
Determine the probability of the given opposite event. What is the probability of rolling a fair die and not getting an outcome less than 3?
The probability of rolling a fair die and not getting an outcome less than 3 is 2/3
Determine the probability of the given opposite event. What is the probability of rolling a fair die and not getting an outcome less than 5?
The probability of rolling a fair die and not getting an outcome less than 5 is 1/3.
You toss a coin 300 times and get 45 heads. Complete parts (a)-(c) below.
-Determine the relative frequency probability. 0.15 -Determine the expected frequency of the event. 0.50 -Yes, because the relative frequency is not near the expected frequency.
You toss a coin 100 times and get 14 heads. Complete parts (a)-(c) below.
-Determine the relative frequency probability. 0.14 -Determine the expected frequency of the event. 0.5 -Do you have reason to suspect the coin is unfair? Explain. Yes, because the relative frequency is not near the expected frequency.
Explain the meaning of "life expectancy." How does life expectancy change with age? How is it affected by changes in the overall health of a population?
-Life expectancy is the number of additional years a person of a given age can expect to live on average. -Decreases -Increases, Decreases
Decide which method (theoretical, relative frequency, or subjective) is appropriate, and compute or estimate the following probability. Randomly meeting someone with a phone number that ends in 0, 5, 8.
-The theoretical method should be used. -The probability of randomly meeting someone with a phone number ending in 0,5, or 8 is 3/10.
Decide which method (theoretical, relative frequency, or subjective) is appropriate, and compute or estimate the following probability. Randomly meeting someone with a phone number that ends in
-The theoretical method should be used. -The probability of randomly meeting someone with a phone number ending in 3 is 1/10.
Suppose you toss a fair coin 10,000 times. Should you expect to get exactly 5000 heads? Why or why not? What does the law of large numbers tell you about the results you are likely to get?
-You shouldn't expect to get exactly 5000 heads, because you cannot predict precisely how many heads will occur. -The proportion of heads should approach 0.5 as the number of tosses increases.
The table shows the leading causes of death in a certain country in a recent year. The population of the country was 318 million. What is the empirical probability of death by heart disease during a single year? How much greater is the risk of death by heart disease than death by suicide?
.00188 (.001877 calculator) 15.6 (15.584 on calculator)
Getting rain at least once in 10 days if the probability of rain on each single day is 0.5.
.999
If we are only interested in the number of W days, what are the possible events for two consecutive days.
0,1,2
Determine the probability of the given opposite event. What is the probability that a 44% free-throw shooter will miss her next free throw?
0.56 (44% of 100 = 0.44. 0.44-1= 0.56)
Meeting at least one left-handed person in five random encounters on campus when the incidence rate is 17% (17 in 100 people are left-handed)
0.606
Pizza House offers 5 different salads, 5 different kinds of pizza, and 3 different desserts. How many different three course meals can be ordered?
75
How many different choices of car does a person have if a particular model comes in 16 colors and 5 styles (hatchback, mid-size SUV, minivan, sedan, or station wagon)?
80
The table shows the leading causes of death in a certain country in a recent year. The population of the country was 313 million. If you lived in a typical city of 500,000, how many people would you expect to die of heart disease each year?
909
What is a probability distribution? Explain how to make a table of a probability distribution.
A probability distribution represents the probabilities of all possible events of interest.
Choose the best answer to the following question. Explain your reasoning with one or more complete sentences. A box contains 20 chocolates, but only two of them are the dark chocolate that somebody likes. When it is said that the probability of drawing one of the dark chocolates at random is 0.1, what kind of probability is being stated?
A theoretical probability, because both the total and desired numbers of objects are known
Explain the meaning of the law of large numbers. Does this law say anything about what will happen in a single observation or experiment? Why or why not?
As the experiment is done more and more times, the proportion of times that a certain outcome occurs should get closer to the theoretical probability that that outcome would occur.
Use the theoretical method to determine the probability of the following outcome and event. State any assumptions made. Tossing two coins and getting either one head or two heads.
Assuming that each coin is fair and is equally likely to land heads or tails, the probability is 3/4.
Randomly selecting a four-person committee consisting entirely of men from a pool of 10 men and 14 women
Dependent .0198
Drawing three fives in a row from a standard deck of cards when the drawn card is not returned to the deck each time
Dependent 1/5525
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Drawing three face cards in a row from a standard deck of cards when the drawn card is not returned to the deck each time
Dependent 11/1105
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Randomly selecting a four-person committee consisting entirely of women from a pool of 18 men and 10 women
Dependent 0.0103
Drawing three non-face cards in a row from a standard deck of cards when the drawn card is not returned to the deck each time
Dependent 38/85
How is the expected value EV of two events computed?
EV = (event 1 value) x (event 1 probability) + (event 2 value) x (event 2 probability)
What is an expected value and how is it computed? Should we always expect to get the expected value? Why or why not?
Expected value is the estimated gain or loss of partaking in an event many times.
When one coin is tossed, the probability of landing heads is 1/2. Assuming the coin is fair, what does that statement mean?
If 1000 coins are tossed, there is no way to predict the precise number of heads that will be generated, though it will probably be close to 500 because probability does not guarantee certain results.
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. The next ten births at a hospital all being girls.
Independent 1/1024
The next seven births at a hospital all being girls.
Independent 1/128
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. The next six births at a hospital all being boys.
Independent 1/64
A $1 slot machine at a casino is set so that it returns 97% of all the money put into it in the form of winnings, with most of the winning in the form of huge but low-probability jackpots. What is your probability of winning when you put $1 into this slot machine?
It cannot be calculated from the given data, but it is certainly quite low. We would need to know the value of the jackpots to determine the correct probability.
Drawing either a non-face card or a queen from a regular deck of cards
Non overlapping
Drawing either a 2 or a queen from a regular deck of cards.
Non overlapping 2/13
Drawing either a black seven or a black three on one draw from a regular deck of cards.
Non-overlapping 1/13
Which of the following is true for the possible range of values for P(A)?
The range of possible values for P(A) is from 0 to 1 (inclusive), with 0 meaning there is no chance that event A will occur and 1 meaning it is certain that event A will occur.
Suppose that the probability of a hurricane striking a state in any single year is 1 in 10 and that this probability has been the same for the past 1000 years. Which of the following is implied by the law of large numbers?
The state has been hit by close to (but not necessarily exactly) 100 hurricanes in the past 1000 years because as the years increase, the closer the proportion of hurricanes should be to 0.1.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. Your life expectancy is the major factor in determining how long you live.
The statement does not make sense because life expectancy is based on the current state of medicine and technology. If you have healthy habits and exercise each day, then you are more likely to live longer, regardless of your life expectancy.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I haven't won in my last 25 pulls on the slot machine, so I must be having a bad day and I'm sure to lose if I play again.
The statement does not make sense because the results of repeated trials do not depend on results of earlier trials.
Decide whether the following statement makes sense (or clearly true) or does not make sense (or is clearly false). Explain your reasoning. The probability that Jonas will win the race is 0.6 and the probability that he will not win is 0.5.
The statement does not make sense because the sum of the probabilities of Jonas winning and not winning the race must equal to 1.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The probability of drawing an ace or a spade from a deck of cards is the same as the probability of drawing the ace of spades.
The statement does not make sense because there is one card that is the ace of spades but more than one card that is either an ace or a spade.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The expected value to me of each raffle ticket I purchased is −$0.85.
The statement makes sense because a negative expected value implies that, averaged over many tickets, you should expect to lose $0.85 for each raffle ticket that you buy.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The probability of getting heads and tails when you toss a coin is 0, but the probability of getting heads or tails is 1.
The statement makes sense because heads and tails are the only possible outcomes and it is impossible to get both heads and tails on a single coin toss.
Decide whether the following statment makes sense (is clearly true) or does not make sense (is clearly false). Explain your reasoning. A 60-year-old has a shorter life expectancy than a 20-year-old.
The statement makes sense because life expectancy measures the number of additional years a person of a given age can expect to live on average. Therefore, someone who is 20 has higher life expectancy than someone who is 60.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. When I toss four coins, there are four different outcomes that all represent the event of three heads and one tail.
The statement makes sense. There are four different ways the event can occur.
A group of friends are playing 5-card poker with a deck of 52 cards. For a probability distribution showing the individual probabilities of all possible hands, what would be the sum of all the individual probabilities?
The sum would equal 1 because the sum of the probabilities of all possible events in any situation is 1.
Briefly describe the differences among theoretical, relative frequency, and subjective techniques for finding probabilities. Give an example of each.
The theoretical technique is based on the assumption that all outcomes are equally likely, while the relative frequency technique is based on observations or experiments, and the subjective technique is an estimate based on experience or intuition. -Which one of the following is an example of a theoretical probability? The probability of rolling a 3 on a single die is 1/6. -Which one of the following is an example of a relative frequency probability? Based on statistical data, the chance of having the championship team coming from the Eastern Conference of a certain basketball league is about 1 in 10. -Which one of the following is an example of a subjective probability? My teacher assures me that he is certain that my SAT scores will be the highest for the entire country.
Explain why the probability is the same for any particular set of ten coin toss outcomes. How does this idea affect our thinking about streaks?
The total number of outcomes for ten coins is 2×2×2×2×2×2×2×2×2×2=1024, so every individual outcome has the same probability of 1/1024. A streak of all heads would not seem surprising since a streak of all heads is just as likely as a streak of all tails, or as likely as any other combination of outcomes.
Which of the following correctly describes the events of being born on a Wednesday and being born in July?
The two events are overlapping. It is possible to be born on a Wednesday in July.
Consider a lottery game in which six balls are drawn randomly from a set of balls numbered 1 through 42. One week, the winning combination consists of balls numbered 5, 12, 23, 32, 36, and 41. The next week, the winning balls are numbered 1, 2, 3, 4, 5, and 6. Is the second winning set more or less likely than or just as likely as the first? Explain.
They are equally likely because every combination of six balls is equally likely.
My chance of getting a 5 on the roll of one die is 1/6, so my chance of getting at least one 5 when I roll three dice is 3/6.
This does not make sense because the real probability would be 1−563, which is not equal to 3/6.
Does this law say anything about what will happen in a single observation or experiment? Why or why not?
This law does not say anything about what will happen in a single observation or experiment. Large numbers of events may show some pattern, but the individual events are unpredictable.
Cameron is betting on a game in which the probability of winning is 1 in 5. He's lost five games in a row, so he decides to double his bet on the sixth game. What does this strategy show?
This shows poor logic, as he has an 80% chance of losing the double bet. The past bad luck has no bearing on the future chances. His chances of winning never change.
Explain how to make a table of a probability distribution. Choose the correct answer below.
To make a table of a probability distribution, list all possible outcomes, identify the outcomes that represent the same event, and then find the probability of each event.
What are vital statistics? How are they usually described? Give a few examples.
Vital statistics are data concerning births, deaths, and life expectancy of citizens. They are usually expressed in terms of deaths per person or per 100,000 people or as the number of years a person is expected to live. Two examples are the rate of deaths caused by cancer and the rate of deaths caused by heart disease.
Suppose we describe the weather as either warm (W) or chilly (C). Answer parts (a) and (b) below. a. Using the letters W and C, list all the possible outcomes for the weather on two consecutive days.
WC, WW, CC, CW
Should we always expect to get the expected value? Why or why not?
We should not always expect to get the expected value because expected value is calculated with the assumption that the law of large numbers will come into play.
Consider a lottery with 100 million tickets in which each ticket has a unique number. Each ticket is sold for $1, and one ticket is drawn for a single prize of $75 million (and no other prizes). If you were to spend $1 million to purchase 1 million lottery tickets, what would the most likely result be?
You would lose your entire $1 million. No matter how many tickets are bought, the chance of winning the lottery is still 1 in 100 million.
Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Drawing three face cards in a row from a standard deck of cards when the drawn card is not returned to the deck each time
dependent 11/1105
Determine whether the following individual events are overlapping or non-overlapping. Then find the probability of the combined event. Drawing either a club or a red card from a regular deck of cards
non-overlapping 3/4
