at a particular carnival, there is a dice game that costs $5 to play. -if the die lands on an odd...

At a particular carnival, there is a dice game that costs$5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $14.

How much could I “walk away with” for each of the possible outcomes?

$5

How much money did you win/get back?

- I did not get any of the money back


-Lands on an odd number:How much did you pay?

Did you walk away with more or less $?

- I walk away losing $5

$5


- I got back $8


-Lands on a 2 or 4:How much did you pay?


- I walk away with $3 more than I started

$5


- I got back $13


-Lands on a 6:How much did you pay?



The overall amount you “walk away with”(positive or negative) is called the:

Did you walk away with more or less $?I walk away with $8 more than I started?

Net Gain

There is a game at the fair where you pay $10 to flip aCoin once-If the coin lands heads up, you lose.-If the coin lands tails up, you win $19


$10


- I did not get any of the money back


-Lands heads up:How much did you pay?


- I walk away losing $10

$10


- I got back $19


-Lands tails up:How much did you pay?



What is your net gain if you lose?

If you lose, the net gain = -10

What is your net gain if you win?

If you win, the net gain = 9

Have you ever wondered……..

When playing a game, your chancesMay seem good, but do you thinkThat the odds are in your favor?

Anything deal with chance suchSuch as a casino or lottery….

What does a business have to do to In order to be successful?

Therefore….

At the end of the day, the “business”Will have a positive net gain and the“players” will have an overall Negative net gain

Back to our dice example…..At a particular carnival, there is a dice game that costs$5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $13.

Now we can create a probabilityDistribution with out possible

Outcomes and our net gains

What could be your possible “winnings”?

At a particular carnival, there is a dice game that costs$5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $13.

Lose, win $8, win $13

“Winnings” Lose Win $8 Win $13

Net Gain -5 3 8

P(X) 3/6 2/6 1/6

Mean = -5 (3/6) + 3 (2/6) + 8 (1/6)

Now find the “mean” using net gain and P(X)

Mean = -2.5 + 1 + 1.33

Mean = -0.2

This “mean” we found using net gain and P(X)Is called:

Expected Value = E(X)

Therefore, each time I play the dice game I am Expected to lose $0.20 on average.

Does this seem correct that I expect to lose?

Yes, because that means the “business” is making $

Find the expected value from our coin exampleThere is a game at the fair where you pay $10 to flip aCoin once-If the coin lands heads up, you lose.-If the coin lands tails up, you win $19

“Winnings” Lose Win $8

Net Gain -10 9

P(X) 1/2 1/2

E(X) = -10 (1/2) + 9 (1/2)E(X) = -5 + 4.5 = -0.5

Example 1:Find the expected value if tickets are sold in a raffle at$2 each. The prize is a $1000 shopping spree at a localMall. Assume that one ticket is purchased.

“Winnings” Lose Win

Net Gain -2 998

P(X) 14991500

E(X) = -2(1499/1500)+ 998(1/1500)E(X) = -1.999 + 0.665 = -1.33

_1__1500

Example 2:Find the expected value for example #1 if two tickets

Are purchased

“Winnings” Lose Win

Net Gain -4 996

P(X) 14981500

E(X) = -4(1498/1500)+ 996(2/1500)E(X) = -3.995 + 1.328 = -2.67

_2__1500

Example 3:A lottery offers one $1000 prize, one $500 prize, andFive $100 prizes. One thousand tickets are sold at $3

each. Find the expected value of one ticket.


Net Gain -3 997

P(X) 993_1000

E(X) = -3(993/1000)+ 997(1/1000) + 497(1/1000) + 97(5/1000)

E(X) = -2.979 + 0.997 + 0.497 + 0.485 = -1.00

_1__1000

Win $500

497_1__1000

Win $100

97_5__1000

Try some on your own:One thousand tickets were sold at $1 each for four

Prizes of $100, $50, $25, and $10. What is the Expected value if a person purchases two tickets?


Net Gain -2 98

P(X) 992_1000

E(X) = -1.63

_2__1000

Win $50

48_2__1000

Win $25

23_2__1000

Win $10

8_2__1000

Try some on your own:You pay $5 to draw a card from a standard deck of 52Cards. If you pick a red card, you win nothing. If you

Get a spade, you win $5. If you get a club, you win $10.If you get the ace of clubs, you win an additional $20.

Find the expected value of drawing one card.

“Winnings” Red Spade

Net Gain -5 0

P(X) 2652

E(X) = -0.87

1352

Club

51252

Ace of Clubs

251_52

at a particular carnival, there is a dice game that costs $5 to play. -if the die lands on an odd...

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