Expectation Exercises.

Expectation Exercises.

Pages 380 ? 382

373

Problems 1,4,5,7 (you don't need to use trees, if you don't want to...but they might help!), 9,11-15

1 ? 5 (you'll need to head to this page: )

E1. 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 $1). If not, you lose the $1. 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 a single number... 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 $1)?

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 platforma.

1. 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 ).

E7. Some of you may have seen the TV show "Deal or No Deal" (DOND). It's a nice extension of both the Monty Hall problem and the idea of expectation averages for probabilities. In case you've never played, try this! )

Now, if you're like me, you might wonder "How the does the banker get that offer?" Try this out!



Is there a simpler formula?

Answers.

E1. Since there are 38 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 38. The event space is 18 (since there are 18 red spaces). So,

18

20

there's an 38 chance you win $1, and a 38 chance you lose $1. Using the definition of expectation, you can expect to

18

20

win 38($1) + 38(-$1), or about -$0.05. Thus, you'll lose, on average, 5 cents per game.

But how can that be? You either win $1, or lose $1. How can you average a 5 cent loss?

E2. You've got this!

E3. Remember...there are 38 spaces, and you're picking only 1 upon which to place your bet.

E4. Hint: they're "odd", "Even", "High" and "Low". Have fun!

E5. Mine took about 1000(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?

E7. Probably.

Expectation Quizzes.

Quiz 1.

In class, we calculated the probabilities of getting various sums when you roll a pair of dice (if you've ever played Catan, they come in handy). We also played craps, the perennially favorite game of chance, where you can't help but lose your money.

Let's create some new games here. Let's suppose you make a up a game that involves getting "5" on a pair of dice; the more 5's you get, the better (so, getting double 5's would be the big winner).

Here's an image (that we probably used in class) of all ways 2 dice can roll, with all the ways a "5" can show up on at least one of the rolls:

Here'a a more tabular way of looking at these results!

X = number of

seen on two dice!

0 1 2

Chance of that many 5's:

25 36 69% 10 36 28%

1 36 3%

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 $1 on this game:

X = number of Payout

seen

What this translates to (in $)!

chance

0

0:1

-$1 (you lost your bet of $1)

69%

1

1:1 $0 (you won back your "ante" ? that is, your bet) 28%

2

2:1

$1 (you won back $2, so profited $1)

3%

So, on average, here's how much you would earn, over time, when you bet $1 each time:

(weighted) Average winnings

=

-$1*0.69 + 0.69

$0*0.28 + $1*0.03 + 0.28 + 0.03

=

-$.67 1

= -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 $1. You have them a $1 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

Payout

What this translates to (in $)!

P(X)

seen

0

0:1

-$1

69%

1

1:1

$0

28%

2

5:1

$4 (you won back $5, so profited $4)

3%

-$1*0.69 + $0*0.28 + $4*0.03 -$.58

(weighted) Average winnings =

0.69 + 0.28 + 0.03

= 1 = -58?

1. (2 points) Will Vegas like this one? Why or why not?

2. (3 points) What are your average winnings if Vegas offers you a 10:1 payout on the "2 fives" option?

3. (3 points) What are your average winnings if Vegas offers you a 10:1 payout on the "2 fives" option AND a 2:1 payout on the "1 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 0 1

2

seen Odds Paid 0:1 1:1

?:1

Quiz 2. 1. (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 $1b

on such a bet. Suppose, additionally, that the payout is 2:1 (that is, you earn back 2 times what you bet if you win...so, for a $1 bet, you'd win back $2, but profit $1). 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 36:1 (!). 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 $1 on green for up to 37 games ? since the chance of hitting on green is 1/37, 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 1.

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