Expectation Exercises.

Transcription

1 Expectation Exercises. Pages Problems 0 2,4,5,7 (you don t need to use trees, if you don t want to but they might help!), 9, (you ll need to head to this page: E. In the casino game of roulette, a wheel is spun, and a little ball drops into one of many numbered spots (here s a video if you need a visual: The simplest way to bet is either on red or black. Let s say you bet ( ante up ) a dollar on red. If the wheel is spun, and it comes up red, you win $2 (which means you profited $). If not, you lose the $. What can you expect to earn, over time, if you bet this way on an American roulette wheel? That s one of the American types shown at right. E2. Assuming the same bet/payout scheme, what would you expect to earn, over time, on a European Roulette wheel (shown at left)? E3. Now imagine betting (on the American roulette wheel) on, say, the number 28. If you hit this number, you are paid out at 35 times your bet amount. What are your expected earnings on this bet (say, again, you bet $)? E4. Think back to class, when we played Chuck A Luck. There are other ways to bet (other than simply picking one number). Analyze their payouts. Refer back to the rad Plinko generator website above. Set up the simulator to look like the one at right (make sure to set the one option to fraction, which will give you the experimental probabilities associated with each outcome; also, I have the rows set to 5). This experiment models the situation of how many business, out of a random sample of 5, will adopt the newest Windows operating platform a.. Go ahead and hit Start and let it run until the graph seems to stabilize. 2. Then press Stop. E5. About how many trials did it take to roughly stabilize? This number is found in the upper left corner ( N= ). E6. On average, how many businesses (out of 5) will adopt the newest windows platform? a

2 E7. The following file in parentheses ( is a file we sometimes use in class to begin to gather data to explore exactly what the banker does when he gives you an offer in the game Deal Or No Deal (DOND). a. Open the sheet 6DOND.xls. b. Open the DOND play online page ( c. Click Start Game. d. You ll need your TI (or some other random number generator). For your TI, type the sequence of buttons to help you be random. Press to select your case. Whatever case the TI tells you, click on that. e. You re now going to play the game 9 times, choosing no deal at every turn. The computer game will tell you how many cases you have to open, and the TI will decide which cases to open. Each time you press the TI will generate a new random number between and 26. Sometimes, the TI will give you a number of a case that s already opened (or your case s number). If that happens, just press again to select a number that hasn t been accounted for. f. Each time the banker gives you an offer, write it in the space in the P column. g. Then, adjust the grid in columns B through H to let Excel know which cases have been opened (remove the x next to the $ amount). h. Lastly, write the expected value of the remaining cases in the Q column. Below, I have a screenshot of what happened after I played the first round of this game: i. Press the red No Deal button. j. Keep opening briefcases, writing down banker s offers, and keeping track of expected values until you re forced to open your own case (this should be nine banker offers altogether). k. Average your results. Answers. E. Since there are total spots for the ball to lend, and they re all equally likely to occur (since they re the same size and shape), we can say that the sample space is. The event space is 8 (since there are 8 red spaces). So, there s an 8 20 chance you win $, and a chance you lose $. Using the definition of expectation, you can expect to win 8 20 ($) + (-$), or about -$0.05. Thus, you ll lose, on average, 5 cents per game.

3 But how can that be? You either win $, or lose $. How can you average a 5 cent loss? E2. You ve got this! E3. Remember there are spaces, and you re picking only upon which to place your bet. E4. Hint: they re odd, Even, High and Low. Have fun! E5. Mine took about 000(ish). E6. You can estimate the probabilities by using the vertical axis of the graph. The payouts are just the numbers 0 through 5. BTW how does the value you get compare to the in the upper left corner?

4 Expectation Quizzes. Quiz. In class, we calculated the probability of the various scenarios of betting on a number (we said 5 ) and keeping track of how many times we saw that number come up on a pair of dice. Here s what we arrived at: X = number of 0 2 seen P(X) 5 6 *5 6 = % 6 * * 6 = % 6 * 6 = 36 3% If you need another way to visualize those fractions (the out of 36 ones), try this graphic: See the ratios now? So, now let s play Vegas. In order for this game to be appealing to players, it needs to pay out perceivably big amounts when you hit the double 5, but, to be appealing to Vegas, it needs to collect money on average for the casino. Let s try one possible payout scenario on a bet of $ on this game: X = number of seen Odds Paid Payouts P(X) 0 0: -$ (you lost your bet of $) 69% : $0 (you won back your ante that is, your bet) 28% 2 2: $ (you won back $2, so profited $) 3% So, on average, here s how much you would earn, over time, when you bet $ each time:

5 (weighted) Average winnings = -$* $0* $* = -$.67 = -67 Well, Vegas ll love this one! On average, they make 67 cents off of you (since you lose 67 cents each time you play, on average that s what the negative means). Think of it this way you go to this table to make change for $. You have them a $ bill, and they hand you back 33 cents. Not bad for them! Ok, let s try another payout scheme that looks a little better to the player. Will Vegas still like it? X = number of seen Odds Paid Payouts P(X) 0 0: -$ (you lost your bet of $) 69% : $0 (you won back your ante that is, your bet) 28% 2 5: $4 (you won back $5, so profited $4) 3% (weighted) Average winnings = -$* $0* $4* = -$.58 = -58. (2 points) Will Vegas like this one? Why or why not? 2. (3 points) What are your average winnings if Vegas offers you 0: odds on the 2 fives option? 3. (3 points) What are your average winnings if Vegas offers you 0: odds on the 2 fives option AND 2: odds on the five option? 4. (2 points) Complete the odds paid chart below (in other words, replace the? with a number) so that your average winnings are $0 (this is called the breakeven point for this game and Vegas knows what it is. ) X = number of seen Odds Paid 0 0: : 2?:

6 Quiz 2.. (2 points) Do a little Googling (or looking up ) and find out how a European (or French) roulette wheel is different than an American one. 2. (4 points) (w) Let s assume you bet on red on such a Euro roulette wheel. Suppose, also, that the ante is $ b on such a bet. Suppose, additionally, that the payout is 2: (that is, you earn back 2 times what you bet if you win so, for a $ bet, you d win back $2, but profit $). What would your expected earnings be on this game, to the nearest cent? 3. (4 points) (w) Repeat #2 for a bet on green. The payout for this one is 36: (!). 4. (extra 2 points) (w) I have an idea to help us gamble longer! Since the payout on green is so high, why don t we just do this: take whatever amount of money I m willing to bet (say, for simplicity, $37), and divide it into 37 one dollar chips. Then, bet $ on green for up to 37 games since the chance of hitting on green is /37, once I hit it, I ll win back almost all of my money, and then I can continue this pattern indefinitely. Heck, I might even get lucky and hit it early! What s wrong with this logic? Make sure to support your answer with calculations! b If you re feeling English, you can pretend it s.

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