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Common Core Standards: Math

High School: Statistics and Probability

Using Probability to Make Decisions HSS-MD.B.5

5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

One of the most fun things to do with statistics is to show how games can be played with them. More importantly, how to calculate beforehand whether or not it's even worth playing a game based on the expected value of the game.

Standard Components

Drills

  1. If an unfair coin is tossed into the air and drops without interruption, its probability of landing on heads is 0.51 whereas the probability of landing on tails is 0.49. If the payoff for landing on heads is $0.99 and the payoff for landing on tails is $1.01 and it costs $1 to play each game, should you play?

    Correct Answer:

    No, because over the long run your expected value is less than the cost to play

    Answer Explanation:

    We can find the expected value by using E(x) = x1p1 + x2p2 + … + xipi, where x is associated with the payoffs (99 for heads and 101 for tails) and the p values are the probabilities of each. This gives us E(x) = 99 • 0.51 + 101 • 0.49 = 99.98. Our expected value over the long run is less than the cost to play. We will lose money over time playing this game. Even if though the expected value is greater than the lowest payoff, we'd still be losing money if we played.


  2. A new game consists of choosing a number from 300. If you pick the correct number, you will win $2,500. Of course, the game costs money to play. When would the price be too high for you to play and have a chance of making some money over the long run?

    Correct Answer:

    $9

    Answer Explanation:

    The probability of winning is 1 out of 300 and the payoff is $2,500; that means our expected value is $2,500/300 = $8.33. If the game costs any more than that, we'd expect to lose money in the long run. Since the lowest number higher than $8.33 is $9, (C) is the correct answer.


  3. Your best friend wants to start an Internet company. He believes the probability that it will succeed is quite low, 0.01, but the payoff could be as much as $10 million. You get really excited, but things get even better! Another friend says her idea has a 0.48 chance of succeeding but the payoff is a smaller $28,000. Your male friend asks you invest only 1% of the potential payoff while your female friend asks you invest 85% of the potential payoff. Which friend do you go with to ensure maximal success, assuming an infinite number of re-dos?

    Correct Answer:

    Your female friend

    Answer Explanation:

    You'd have to invest $10 million • 0.01 = $100,000 in your male friend's business and 28,000 • 0.85 = $23,800 in your female friend's business. Out of 100 times, you'd lose 99 ($9,900,000) with your male friend's business for every time you win ($10,000,000), which leaves you with an expected value of $100,000 in the long run—the same you invested. With your female friend, you'd lose 52 times out of 100 ($1,237,600) and you'd win 48 times ($1,344,000). Your net gain would be $106,400, which is a significant improvement from your $23,800 investment. Help your female friend out. (Plus, who has $100,000 lying around for something that has a 1% chance of success?)


  4. If the probability that a basketball team will win is 0.45, what is the maximum amount of money you should bet on the team winning in order to make money over the long run? Assume you pay $1 per game and your payoff is $1.

    Correct Answer:

    $0

    Answer Explanation:

    Clearly over the long run, you will lose all of your money as the probability of winning is less than half and your payoff equals your losses. Basically, you're more likely to lose (55 to 45), and even if you do win, you won't earn anything (because you pay $1 and you'd just earn $1 back).


  5. If you bet $1 on a random outcome that has a 0.6 chance of occurring with a $1 payout for winning, how much money do you expect to make after 100 bets assuming you start with $100?

    Correct Answer:

    $20

    Answer Explanation:

    Since the probability is 0.6 that you will earn $1 and 0.4 that you will lose $1, one hundred bets give you 60 wins and 40 losses. That translates to $60 earned and $40 spent, which means an overall gain of $20.


  6. You pick up a deck of cards that you know is rigged. The deck of 10 cards has 5 cards that are spades, 3 cards that are hearts, and 2 cards that are diamonds. You tell your friend, who has foolishly decided to play against you, that it costs $2.50 to play each time. If he randomly picks a spade, he wins $1, if he picks a heart, he wins $3, if he picks a diamond, he wins $5. What is the expected value of this game after 10 rounds of play?

    Correct Answer:

    -$1

    Answer Explanation:

    You can multiply the earnings of each card by the probabilities to find the expected value of total money earned for every game. That equals 0.5 • $1 + 0.3 • $3 + 0.2 • $5 = $2.4. Since it costs $2.50 to play, the average gain per game is -$0.10. If your friend plays ten games, he's expected to earn $24 and pay $25, meaning a net gain of -$1.


  7. A spin of a casino wheel gives you an equal chance to land on each section. The probability is 1 in 5 that you will do so. However each spot has a different payoff. Find the expected value after one spin.

    Section 1: $1,000

    Section 2: -$3,000

    Section 3: -$2,000

    Section 4: $2,000

    Section 5: $500

    Correct Answer:

    -$300

    Answer Explanation:

    The probability associated with each section is 0.2 We can calculate our expected value by multiplying the earnings by the probability and adding them up (essentially, using E(x) = x1p1 + x2p2 + … + xipi). If we do this, we end up with -$300 as our expected value.


  8. What is the theoretical probability of picking out a milk chocolate Hershey's kiss in a bag that contains 3 milk chocolate Hershey's kisses, 5 white chocolate kisses, and 3 peppermint kisses?

    Correct Answer:

    3/11

    Answer Explanation:

    There are a total of 11 Hershey's kisses in the bag, which is a big indicator already that (B) is right. Also, we're looking for the probability of choosing a milk chocolate one, and there are 3 of those. That means our probability is 3 out of 11, or (B).


  9. When playing on the roulette wheel in casinos, you place your money on red or black numbers. If it lands on the color you chose, the payoff is 1:1 (if you place a $1 bet, you get that bet back plus $1 more). Theoretically, you can use what is called a martingale strategy: if you lose, you simply double down and bet again, and continue to do so until you finally get the color you were looking for. Which of the following is an example of a way that a casino can prevent gamblers from using this strategy?

    Correct Answer:

    Placing a ceiling bet limit so that it's impossible to bet more than a certain amount

    Answer Explanation:

    If the casino institutes a bet limit, you'll eventually reach a ceiling whereby the martingale strategy will no longer work. You won't be able to double down past this limit if you lose a bet. A floor limit won't have this effect since you'll still be able to double. Using hollow roulette balls will have no effect on the betting strategy, and an equal number of red and black spaces just translates to a 50:50 chance of winning or losing.


  10. In a bag are a bunch of pennies. All of them feel the same to you, so your choice is random. Ten of them are from the years 1980 – 1990, fifteen are from 1990 – 2000, and thirteen others are from 2000 – 2010. What is the probability of picking a penny from the year 1993?

    Correct Answer:

    Not enough information

    Answer Explanation:

    While it is in the decade 1990 – 2000, we cannot be sure how many of the 15 are actually from the year 1993. Any number from 0 to 15 of them could be from the year 1993, but there is no way to know. Therefore, there isn't enough information to answer this question.


Aligned Resources

    More standards from High School: Statistics and Probability - Using Probability to Make Decisions