4.4: Expected Value
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Expected value is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it's one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.
Example \(\PageIndex{1}\)
In the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?
Solution
Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you’ve lost $2. On a per-space basis, you have “won” -$2/$38 ≈ -$0.053. In other words, on average you lose 5.3 cents per space you bet on.
We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).
There is another way to compute expected value without imagining what would happen if we play every possible space. There are 38 possible outcomes when the wheel spins, so the probability of winning is \(\dfrac{1}{38}\). The complement, the probability of losing, is \(\dfrac{37}{38}\).
Summarizing these along with the values, we get this table:
|
Outcome |
Probability of Outcome |
|---|---|
|
$35 |
\(\dfrac{1}{38}\) |
|
-$1 |
\(\dfrac{37}{38}\) |
Notice that if we multiply each outcome by its corresponding probability we get \(\$ 35 \cdot \dfrac{1}{38} = 0.9211\) and \(-\$ 1 \cdot \dfrac{37}{38}=-0.9737\), and if we add these numbers we get \(0.9211+(-0.9737) \approx-0.053 \), which is the expected value we computed above.
Expected Value
Expected Value is the average gain or loss of an event if the procedure is repeated many times.
We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products
Try it Now 12
You purchase a raffle ticket to help out a charity. The raffle ticket costs $5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth $4000. Compute the expected value for this raffle.
Example \(\PageIndex{2}\)
In a certain state's lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, then win $1,000. It costs $1 to buy a ticket. Find the expected value.
Solution
Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:
\(\dfrac{_{6} C_{6}}{_{48} C_{6}}=\dfrac{1}{12271512} \approx 0.0000000815 \) for all 6 numbers,
\(\dfrac{\left(_{6} C_{5}\right)\left(_{42} C_{1}\right)}{_{48} C_{6}}=\dfrac{252}{12271512} \approx 0.0000205 \) for 5 numbers.
Our probabilities and outcome values are:
|
Outcome |
Probability of outcome |
|---|---|
|
$999,999 |
\(\dfrac{1}{12271512}\) |
|
$999 |
\(\dfrac{252}{12271512}\) |
|
-$1 |
\(1-\dfrac{253}{12271512}=\dfrac{12271259}{12271512}\) |
The expected value, then is:
\[(\$ 999,999) \cdot \dfrac{1}{12271512}+(\$ 999) \cdot \dfrac{252}{12271512}+(-\$ 1) \cdot \dfrac{12271259}{12271512} \approx-\$ 0.898 \nonumber \]
On average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1.
In general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money. It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the average winnings are positive, it could be the case that most people lose money and one very fortunate individual wins a great deal of money. If the expected value of a game is 0, we call it a fair game, since neither side has an advantage.
Not surprisingly, the expected value for casino games is negative for the player and is positive for the casino. It must be positive, or they would go out of business. Players just need to keep in mind that when they play a game repeatedly, they should expect to lose money because their expected value is negative. That is fine so long as you enjoy playing the game and think it is worth the cost. But it would be wrong to expect to come out ahead.
Try it Now 13
A friend offers to play a game, in which you roll 3 standard 6-sided dice. If all the dice roll different values, you give him $1. If any two dice match values, you get $2. What is the expected value of this game? Would you play?
Expected value also has applications outside of gambling. Expected value is very common in making insurance decisions.
Example \(\PageIndex{3}\)
A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year . An insurance company charges $275 for a life-insurance policy that pays a $100,000 death benefit. What is the expected value for the person buying the insurance?
Solution
The probabilities and outcomes are:
|
Outcome |
Probability of outcome |
|---|---|
|
$100,000-$275=$99,725 |
0.00242 |
|
$275 |
1-0.00242=0.99758 |
The expected value is ($99,725)(0.00242) + (-$275)(0.99758) = -$33.
Not surprisingly, the expected value is negative; the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.
For people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.
Odds and Risks as they relate to Probability
The odds of an event ("odds", always plural) occurring are the probability (e.g . risk ) that this event will occur divided by the probability that the event will not occur. It can also be expressed as the probability that an event will occur divided by "1 minus the probability that the event will occur".
\[\text {Odds of event } = \dfrac{P}{1-P} \nonumber \]
This probability measure is popular in the world of gambling. If we compute the number of people putting money on one horse winning and the number of people putting money on the horse not winning (i.e. putting money on other horses) we can compute the odds of winning. For example among 3100 persons gambling on horses, 100 persons put money on horse "A" to win and 3000 do not (they bet on other horses). The odds of winning are then 1/30 (100/3100 divided by 3000/3100, which can be simplified as 100/3000, or 1 / 30). In fact in gambling the odds of not winning are preferred and expressed as a ratio X/1. In our example, 30/1, or in words "thirty to one". This means that for every dollar that you bet, you will receive 30 if you win.
While odds are not the same as expected value, we often hear both of these terms as they relate to chance and probability. One way to remember the difference is this: odds are a ratio while expected value is a weighted sum as explained above. Odds are very similar to probability, but be aware of the difference. For instance, if you only have 10% chance of winning a game, the odds against you are 9 to 1 while the probability of your not winning would be 9/10 (which, of course, is 90%).
Exercises
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A ball is drawn randomly from a jar that contains 6 red balls, 2 white balls, and 5 yellow balls. Find the probability of the given event.
- A red ball is drawn
- A white ball is drawn
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Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. What is the probability of drawing one slip of paper from the hat at random and getting:
- A consonant
- A vowel
- A group of people were asked if they had run a red light in the last year. 150 responded "yes", and 185 responded "no". Find the probability that if a person is chosen at random, they have run a red light in the last year.
- In a survey, 205 people indicated they prefer cats, 160 indicated they prefer dots, and 40 indicated they don’t enjoy either pet. Find the probability that if a person is chosen at random, they prefer cats.
- Compute the probability of tossing a six-sided die (with sides numbered 1 through 6) and getting a 5.
- Compute the probability of tossing a six-sided die and getting a 7.
- Giving a test to a group of students, the grades and gender are summarized below. If one student was chosen at random, find the probability that the student was female.
|
A |
B |
C |
Total |
|
|---|---|---|---|---|
|
Male |
8 |
18 |
13 |
39 |
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Female |
10 |
4 |
12 |
26 |
|
Total |
18 |
22 |
25 |
65 |
- The table below shows the number of credit cards owned by a group of individuals. If one person was chosen at random, find the probability that the person had no credit cards.
|
Zero |
One |
Two or more |
Total |
|
|---|---|---|---|---|
|
Male |
9 |
5 |
19 |
33 |
|
Female |
18 |
10 |
20 |
48 |
|
Total |
27 |
15 |
39 |
81 |
- Compute the probability of tossing a six-sided die and getting an even number.
- Compute the probability of tossing a six-sided die and getting a number less than 3.
- If you pick one card at random from a standard deck of cards, what is the probability it will be a King?
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If you pick one card at random from a standard deck of cards, what is the probability it will be a Diamond?
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Compute the probability of rolling a 12-sided die and getting a number other than 8.
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If you pick one card at random from a standard deck of cards, what is the probability it is not the Ace of Spades?
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Referring to the grade table from question #7, what is the probability that a student chosen at random did NOT earn a C?
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Referring to the credit card table from question #8, what is the probability that a person chosen at random has at least one credit card?
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A six-sided die is rolled twice. What is the probability of showing a 6 on both rolls?
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A fair coin is flipped twice. What is the probability of showing heads on both flips?
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A die is rolled twice. What is the probability of showing a 5 on the first roll and an even number on the second roll?
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Suppose that 21% of people own dogs. If you pick two people at random, what is the probability that they both own a dog?
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Suppose a jar contains 17 red marbles and 32 blue marbles. If you reach in the jar and pull out 2 marbles at random, find the probability that both are red.
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Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. If you pull out two slips at random, find the probability that both are vowels.
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Bert and Ernie each have a well-shuffled standard deck of 52 cards. They each draw one card from their own deck. Compute the probability that:
- Bert and Ernie both draw an Ace.
- Bert draws an Ace but Ernie does not.
- neither Bert nor Ernie draws an Ace.
- Bert and Ernie both draw a heart.
- Bert gets a card that is not a Jack and Ernie draws a card that is not a heart.
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Bert has a well-shuffled standard deck of 52 cards, from which he draws one card; Ernie has a 12-sided die, which he rolls at the same time Bert draws a card. Compute the probability that:
- Bert gets a Jack and Ernie rolls a five.
- Bert gets a heart and Ernie rolls a number less than six.
- Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
- Bert gets a red card and Ernie rolls a fifteen.
- Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
- Compute the probability of drawing a King from a deck of cards and then drawing a Queen.
- Compute the probability of drawing two spades from a deck of cards.
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A math class consists of 25 students, 14 female and 11 male. Two students are selected at random to participate in a probability experiment. Compute the probability that
- a male is selected, then a female.
- a female is selected, then a male.
- two males are selected.
- two females are selected.
- no males are selected.
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A math class consists of 25 students, 14 female and 11 male. Three students are selected at random to participate in a probability experiment. Compute the probability that
- a male is selected, then two females.
- a female is selected, then two males.
- two females are selected, then one male.
- three males are selected.
- three females are selected.
- Giving a test to a group of students, the grades and gender are summarized below. If one student was chosen at random, find the probability that the student was female and earned an A.
|
A |
B |
C |
Total |
|
|---|---|---|---|---|
|
Male |
8 |
18 |
13 |
33 |
|
Female |
10 |
4 |
12 |
48 |
|
Total |
18 |
22 |
25 |
81 |
- The table below shows the number of credit cards owned by a group of individuals. If one person was chosen at random, find the probability that the person was male and had two or more credit cards.
|
Zero |
One |
Two or more |
Total |
|
|---|---|---|---|---|
|
Male |
9 |
5 |
19 |
33 |
|
Female |
18 |
10 |
20 |
48 |
|
Total |
27 |
15 |
39 |
81 |
- A jar contains 6 red marbles numbered 1 to 6 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is red or odd-numbered.
- A jar contains 4 red marbles numbered 1 to 4 and 10 blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability the marble is blue or even-numbered.
- Referring to the table from #29, find the probability that a student chosen at random is female or earned a B.
- Referring to the table from #30, find the probability that a person chosen at random is male or has no credit cards.
- Compute the probability of drawing the King of hearts or a Queen from a deck of cards.
- Compute the probability of drawing a King or a heart from a deck of cards.
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A jar contains 5 red marbles numbered 1 to 5 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is
- Even-numbered given that the marble is red.
- Red given that the marble is even-numbered.
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A jar contains 4 red marbles numbered 1 to 4 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is
- Odd-numbered given that the marble is blue.
- Blue given that the marble is odd-numbered.
- Compute the probability of flipping a coin and getting heads, given that the previous flip was tails.
- Find the probability of rolling a “1” on a fair die, given that the last 3 rolls were all ones.
- Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student speaks French, given that the student is female.
- Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
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A certain virus infects one in every 400 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 10% of the time if the person does not have the virus. Let A be the event "the person is infected" and B be the event "the person tests positive".
- Find the probability that a person has the virus given that they have tested positive, i.e. find P(A | B).
- Find the probability that a person does not have the virus given that they test negative, i.e. find P(not A | not B).
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A certain virus infects one in every 2000 people. A test used to detect the virus in a person is positive 96% of the time if the person has the virus and 4% of the time if the person does not have the virus. Let A be the event "the person is infected" and B be the event "the person tests positive".
- Find the probability that a person has the virus given that they have tested positive, i.e. find P(A | B).
- Find the probability that a person does not have the virus given that they test negative, i.e. find P(not A | not B).
- A certain disease has an incidence rate of 0.3%. If the false negative rate is 6% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.
- A certain disease has an incidence rate of 0.1%. If the false negative rate is 8% and the false positive rate is 3%, compute the probability that a person who tests positive actually has the disease.
- A certain group of symptom-free women between the ages of 40 and 50 are randomly selected to participate in mammography screening. The incidence rate of breast cancer among such women is 0.8%. The false negative rate for the mammogram is 10%. The false positive rate is 7%. If a the mammogram results for a particular woman are positive (indicating that she has breast cancer), what is the probability that she actually has breast cancer?
- About 0.01% of men with no known risk behavior are infected with HIV. The false negative rate for the standard HIV test 0.01% and the false positive rate is also 0.01%. If a randomly selected man with no known risk behavior tests positive for HIV, what is the probability that he is actually infected with HIV?
- A boy owns 2 pairs of pants, 3 shirts, 8 ties, and 2 jackets. How many different outfits can he wear to school if he must wear one of each item?
- At a restaurant you can choose from 3 appetizers, 8 entrees, and 2 desserts. How many different three-course meals can you have?
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How many three-letter "words" can be made from 4 letters "FGHI" if
- repetition of letters is allowed
- repetition of letters is not allowed
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How many four-letter "words" can be made from 6 letters "AEBWDP" if
- repetition of letters is allowed
- repetition of letters is not allowed
- All of the license plates in a particular state feature three letters followed by three digits (e.g. ABC 123). How many different license plate numbers are available to the state's Department of Motor Vehicles?
- A computer password must be eight characters long. How many passwords are possible if only the 26 letters of the alphabet are allowed?
- A pianist plans to play 4 pieces at a recital. In how many ways can she arrange these pieces in the program?
- In how many ways can first, second, and third prizes be awarded in a contest with 210 contestants?
- Seven Olympic sprinters are eligible to compete in the 4 x 100 m relay race for the USA Olympic team. How many four-person relay teams can be selected from among the seven athletes?
- A computer user has downloaded 25 songs using an online file-sharing program and wants to create a CD-R with ten songs to use in his portable CD player. If the order that the songs are placed on the CD-R is important to him, how many different CD-Rs could he make from the 25 songs available to him?
- In western music, an octave is divided into 12 pitches. For the film Close Encounters of the Third Kind, director Steven Spielberg asked composer John Williams to write a five-note theme, which aliens would use to communicate with people on Earth. Disregarding rhythm and octave changes, how many five-note themes are possible if no note is repeated?
- In the early twentieth century, proponents of the Second Viennese School of musical composition (including Arnold Schönberg, Anton Webern and Alban Berg) devised the twelve-tone technique, which utilized a tone row consisting of all 12 pitches from the chromatic scale in any order, but with not pitches repeated in the row. Disregarding rhythm and octave changes, how many tone rows are possible?
- In how many ways can 4 pizza toppings be chosen from 12 available toppings?
- At a baby shower 17 guests are in attendance and 5 of them are randomly selected to receive a door prize. If all 5 prizes are identical, in how many ways can the prizes be awarded?
- In the 6/50 lottery game, a player picks six numbers from 1 to 50. How many different choices does the player have if order doesn’t matter?
- In a lottery daily game, a player picks three numbers from 0 to 9. How many different choices does the player have if order doesn’t matter?
- A jury pool consists of 27 people. How many different ways can 11 people be chosen to serve on a jury and one additional person be chosen to serve as the jury foreman?
- The United States Senate Committee on Commerce, Science, and Transportation consists of 23 members, 12 Republicans and 11 Democrats. The Surface Transportation and Merchant Marine Subcommittee consists of 8 Republicans and 7 Democrats. How many ways can members of the Subcommittee be chosen from the Committee?
- You own 16 CDs. You want to randomly arrange 5 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?
- A jury pool consists of 27 people, 14 men and 13 women. Compute the probability that a randomly selected jury of 12 people is all male.
- In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match those drawn, they player wins second prize. What is the probability of winning this prize?
- In a lottery game, a player picks six numbers from 1 to 48. If 4 of the 6 numbers match those drawn, they player wins third prize. What is the probability of winning this prize?
- Compute the probability that a 5-card poker hand is dealt to you that contains all hearts.
- Compute the probability that a 5-card poker hand is dealt to you that contains four Aces.
- A bag contains 3 gold marbles, 6 silver marbles, and 28 black marbles. Someone offers to play this game: You randomly select on marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1. What is your expected value if you play this game?
- A friend devises a game that is played by rolling a single six-sided die once. If you roll a 6, he pays you $3; if you roll a 5, he pays you nothing; if you roll a number less than 5, you pay him $1. Compute the expected value for this game. Should you play this game?
- In a lottery game, a player picks six numbers from 1 to 23. If the player matches all six numbers, they win 30,000 dollars. Otherwise, they lose $1. Find the expected value of this game.
- A game is played by picking two cards from a deck. If they are the same value, then you win $5, otherwise you lose $1. What is the expected value of this game?
- A company estimates that 0.7% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company's expected value of each warranty sold?
- An insurance company estimates the probability of an earthquake in the next year to be 0.0013. The average damage done by an earthquake it estimates to be $60,000. If the company offers earthquake insurance for $100, what is their expected value of the policy?
Some of these questions were adapted from puzzles at mindyourdecisions.com.
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A small college has been accused of gender bias in its admissions to graduate programs.
- Out of 500 men who applied, 255 were accepted. Out of 700 women who applied, 240 were accepted. Find the acceptance rate for each gender. Does this suggest bias?
- The college then looked at each of the two departments with graduate programs, and found the data below. Compute the acceptance rate within each department by gender. Does this suggest bias?
- Looking at our results from Parts a and b, what can you conclude? Is there gender bias in this college’s admissions? If so, in which direction?
|
Department |
Men |
Women |
||
|
Applied |
Admitted |
Applied |
Admitted |
|
|
Dept A |
400 |
240 |
100 |
90 |
|
Dept B |
100 |
15 |
600 |
150 |
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A bet on “black” in Roulette has a probability of 18/38 of winning. If you win, you double your money. You can bet anywhere from $1 to $100 on each spin.
- Suppose you have $10, and are going to play until you go broke or have $20. What is your best strategy for playing?
- Suppose you have $10, and are going to play until you go broke or have $30. What is your best strategy for playing?
- Your friend proposes a game: You flip a coin. If it’s heads, you win $1. If it’s tails, you lose $1. However, you are worried the coin might not be fair coin. How could you change the game to make the game fair, without replacing the coin?
- Fifty people are in a line. The first person in the line to have a birthday matching someone in front of them will win a prize. Of course, this means the first person in the line has no chance of winning. Which person has the highest likelihood of winning?
- Three people put their names in a hat, then each draw a name, as part of a randomized gift exchange. What is the probability that no one draws their own name? What about with four people?
-
How many different “words” can be formed by using all the letters of each of the following words exactly once?
- “ALICE”
- “APPLE”
-
How many different “words” can be formed by using all the letters of each of the following words exactly once?
- “TRUMPS”
- “TEETER”
- The Monty Hall problem is named for the host of the game show Let’s make a Deal . In this game, there would be three doors, behind one of which there was a prize. The contestant was asked to choose one of the doors. Monty Hall would then open one of the other doors to show there was no prize there. The contestant was then asked if they wanted to stay with their original door, or switch to the other unopened door. Is it better to stay or switch, or does it matter?
-
Suppose you have two coins, where one is a fair coin, and the other coin comes up heads 70% of the time. What is the probability you have the fair coin given each of the following outcomes from a series of flips?
- 5 Heads and 0 Tails
- 8 Heads and 3 Tails
- 10 Heads and 10 Tails
- 3 Heads and 8 Tails
-
Suppose you have six coins, where five are fair coins, and one coin comes up heads 80% of the time. What is the probability you have a fair coin given each of the following outcomes from a series of flips?
- 5 Heads and 0 Tails
- 8 Heads and 3 Tails
- 10 Heads and 10 Tails
- 3 Heads and 8 Tails
-
In this problem, we will explore probabilities from a series of events.
- If you flip 20 coins, how many would you expect to come up “heads”, on average? Would you expect every flip of 20 coins to come up with exactly that many heads?
- If you were to flip 20 coins, what would you consider a “usual” result? An “unusual” result?
- Flip 20 coins (or one coin 20 times) and record how many come up “heads”. Repeat this experiment 9 more times. Collect the data from the entire class.
- When flipping 20 coins, what is the theoretic probability of flipping 20 heads?
- Based on the class’s experimental data, what appears to be the probability of flipping 10 heads out of 20 coins?
- The formula \({ }_{n} C_{x} p^{x}(1-p)^{n-x}\) will compute the probability of an event with probability \(p\) occurring \(x\) times out of \(n\), such as flipping \(x\) heads out of \(n\) coins where the probability of heads is \(p = ½\). Use this to compute the theoretic probability of flipping 10 heads out of 20 coins.
- If you were to flip 20 coins, based on the class’ experimental data, what range of values would you consider a “usual” result? What is the combined probability of these results? What would you consider an “unusual” result? What is the combined probability of these results?
- We’ll now consider a simplification of a case from the 1960s. In the area, about 26% of the jury eligible population was black. In the court case, there were 100 men on the juror panel, of which 8 were black. Does this provide evidence of racial bias in jury selection?
- When three coins are tossed, the probability of getting three tails is 1/8. Suppose you get $6 if you get three tails and lose $2 otherwise. Calculate the expected value (this represents your average gain per game).
- A die is tossed in a game. Suppose you win $10 if you get a six and lose $2 otherwise. Calculate the expected value of this game.
- You agree to play this game in which the dealer tosses a die. You lose $2 if you get a one, two, or three. You lose $5 if you get a four or a five. But you win $20 if you get a six. Would you play the game? Answer this by computing the expected value per game.
- Repeat the previous problem, except now that you have to pay $1 to play the game. Would you keep playing?
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A 50-50 raffle is a way to raise money (often used by non-profit organizations). Tickets are sold at an event for a certain price, and the total proceeds are divided evenly (50-50) between one lucky winner and the host organization. Suppose a ticket costs $1, and 700 tickets were sold.
- Find the expected value per ticket purchased. In other words, for each ticket you purchase, how much do you expected to lose/donate?
- Suppose you are the one who bought all 700 tickets sold. How much do you win? Is that amount consistent with your answer for Part (a)?
Reference
- References (9)
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0