7.11.0: Exercises
- Page ID
- 171736
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You roll a standard 6-sided die and win points equal to the square of the number shown.
What’s the expected value of the number of points you win?
Interpret your answer.
In the classic board game The Game of Life, players have the chance to play the market. A spinner with 10 equally likely spaces is spun to choose a random number. If the result is 3 or less, the player loses $25,000. If the result is 7 or more, the player wins $50,000. If the result is a 4, 5, or 6, the player doesn’t win or lose anything.
What is the expected value of playing the market?
Interpret the answer.
The Game of Life players also occasionally have the opportunity to speculate. Players choose any 2 of the 10 numbers on the spinner and then give it a spin. If one of their numbers is chosen, they win $140,000; if not, they lose $10,000.
What is the expected value of this speculation?
Interpret your answer.
Which is better for The Game of Life players: playing the market or speculating? How do you know?
A charitable organization is selling raffle tickets as a fundraiser. They intend to sell 5,000 tickets at $10 each. One ticket will be randomly selected to win the grand prize of a new car worth $35,000.
What is the expected value of a single ticket?
Interpret your answer.
The organization is worried they won’t be able to sell all the tickets, so they announce that, in addition to the grand prize, they will offer 10 second prizes of $500 in cash. What is the new expected value of a single ticket?
Interpret your answer.
In the following exercises involve randomly selecting golf balls from a bucket. The bucket contains 4 yellow balls (numbered 1-4) and 6 white balls (numbered 1-6).
If you draw a single ball, what is the expected number of yellow balls selected?
Suppose you draw 2 balls with replacement.
- Give a PDF table for the possible outcomes for the number of yellow balls selected.
- What is the expected number of yellow balls selected?
Suppose you draw 2 balls without replacement.
- Give a PDF table for the possible outcomes for the number of yellow balls selected.
- What is the expected number of yellow balls selected?
Suppose you draw 3 balls with replacement.
- Give a PDF table for the possible outcomes for the number of yellow balls selected.
- What is the expected number of yellow balls selected?
Suppose you draw 3 balls without replacement.
- Give a PDF table for the possible outcomes for the number of yellow balls selected.
- What is the expected number of yellow balls selected?
If you draw a single ball, what is the expected value of the number on the ball?
Suppose you draw 2 balls with replacement.
- Give a PDF table for the possible outcomes for the sum of the numbers on the selected balls.
- What is the expected sum of the numbers on the balls?
Suppose you draw 2 balls without replacement.
- Give a PDF table for the possible outcomes for the sum of the numbers on the selected balls.
- What is the expected sum of the numbers on the balls?
The following exercises deal with the game “Punch a Bunch,” which appears on the TV game show The Price Is Right. In this game, contestants have a chance to punch through up to 4 paper circles on a board; behind each circle is a card with a dollar amount printed on it. There are 50 of these circles; the dollar amounts are given in this table:
| Dollar Amount | Frequency |
|---|---|
| $25,000 | 1 |
| $10,000 | 2 |
| $5,000 | 4 |
| $2,500 | 8 |
| $1,000 | 10 |
| $500 | 10 |
| $250 | 10 |
| $100 | 5 |
Contestants are shown their selected dollar amounts one at a time, in the order selected. After each is revealed, the contestant is given the option of taking that amount of money or throwing it away in favor of the next amount. (You can watch the game being played in the video Playing "Punch a Bunch.") Anita is playing “Punch a Bunch” and gets 2 punches.
If Anita got $500 on her first punch, what’s the expected value of her second punch?
If Anita got $500 on her first punch, should she throw out her $500 and take the results of her second punch? How do you know?
If Anita got $1,000 on her first punch, what’s the expected value of her second punch?
If Anita got $1,000 on her first punch, should she throw out her $1,000 and take the results of her second punch? How do you know?
If Anita got $2,500 on her first punch, what’s the expected value of her second punch?
If Anita got $2,500 on her first punch, should she throw out her $2,500 and take the results of her second punch? How do you know?
The following exercises are about the casino game roulette. In this game, the dealer spins a marble around a wheel that contains 38 pockets that the marble can fall into. Each pocket has a number (each whole number from 0 to 36, along with a “double zero”) and a color (0 and 00 are both green; the other 36 numbers are evenly divided between black and red). Players make bets on which number (or groups of numbers) they think the marble will land on. The figure shows the layout of the numbers and colors, as well as some of the bets that can be made.
If a player makes a $1 bet on a single number, they win $35 if that number comes up, but lose $1 if it doesn’t. What is the expected value of this bet?
Interpret your answer to the previous question.
Suppose a player makes the $1 bet on a single number in 5 consecutive spins. What is the expected value of this series of bets? (Hint: use the Binomial Distribution.)
Interpret your answer to the previous question.
If a player makes a $10 bet on first dozen, they win $20 if one of the numbers 1–12 comes up but lose $10 otherwise. What is the expected value of this bet?
Interpret your answer to the previous question.
Suppose a player makes the $10 bet on first dozen in 4 consecutive spins. What is the expected value of that series of bets?
Interpret your answer to the previous question.
If a player makes a $10 basket bet, they win $60 if 0, 00, 1, 2, or 3 come up but lose $10 otherwise. What is the expected value of this bet?
Interpret your answer to previous question.
Which is better for the player: a $10 first dozen bet or a $10 basket bet? How do you know?