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Probability & Expected Value

Geometry - Day 57

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AGENDA

  • Check hmwk
  • Do Now
  • New: Expected Value

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Hmwk. p. 951 #8

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Hmwk. p. 951 #9

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Hmwk. p. 951 #10

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Hmwk. p. 951 #11

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Hmwk. p. 951 #12

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Hmwk. p. 951 #13

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Hmwk. p. 951 #14

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Hmwk. p. 951 #14

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Hmwk. p. 951 #15

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Hmwk. p. 951 #15

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Hmwk. p. 951 #16

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Hmwk. p. 951 #16

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Hmwk. p. 951 #17

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Hmwk. p. 951 #18

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Hmwk. p. 951 #19

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Hmwk. p. 951 #19

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Hmwk. p. 951 #20

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Hmwk. p. 951 #20

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Hmwk. p. 951 #29

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Hmwk. p. 951 #29

There are 3 outcomes when 2 or more dice are less than or equal to 4 and 1 outcome when all three are less than or equal to 4. Find the probability of each of the four scenarios and add them together. Answer: 0.74

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Do NOW #1

  1. P(red 1 and sum < 4)

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Do NOW #2

2.) P(blue mult 3 and sum = 8)

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Do NOW #3

3.) Jack and Jill entered a karaoke contest with 12 contestants. What is the probability Jack will finish first and Jill second?

**Does order matter?

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Notes Time

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At the Carnival...

You pay $2 to draw a card

(standard deck).

If you pick an Ace = win!

If you pick a Face = win!

If you pick the 7 of Spades = win!

​

Would you play?

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At the Carnival...

You pay $2 to draw a card

(standard deck).

If you pick an Ace = win!

If you pick a Face = win!

If you pick the 7 of Spades = win!

​

What else do you need to know?

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At the Carnival...

You pay $2 to draw a card

(standard deck).

If you pick an Ace = win $5

If you pick a Face = win $3

If you pick the 7 of Spades = win $10

​

Would you play?

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

​

​

​

​

Prob P(x)

​

​

​

​

Expected Value

​

​

​

​

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

-2+5=3

-2+3+=1

-2+10=8

-2

Prob P(x)

​

​

​

​

Expected Value

​

​

​

​

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

+3

+1

+8

-2

Prob P(x)

4/52

12/52

1/52

35/52

Expected Value

​

​

​

​

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

+3

+1

+8

-2

Prob P(x)

4/52

12/52

1/52

35/52

Expected Value

6/26

6/26

4/26

-35/26

*Sum of EV = What does this mean?

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

+3

+1

+8

-2

Prob P(x)

4/52

12/52

1/52

35/52

Expected Value

6/26

6/26

4/26

-35/26

*Sum of EV = -19/26 or -0.73 : What does this mean?

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

+3

+1

+8

-2

Prob P(x)

4/52

12/52

1/52

35/52

Expected Value

6/26

6/26

4/26

-35/26

*Sum of EV = -19/26 or -0.73

Is it a fair game?

How much will you win/lose if play 100 times?

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At the Carnival...

Let’s organize a table:

​

Event

Ace

Face

7 Spades

Else

Prize (x)

+3

+1

+8

-2

Prob P(x)

4/52

12/52

1/52

35/52

Expected Value

6/26

6/26

4/26

-35/26

*Sum of EV = -19/26 or -0.73

Can you see how they can “rig” a game in their favor?

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Theoretical

Vs.

Experimental Probability

What do you think?

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Theoretical Probability

= What we expect to happen “in theory”.

Ex. Theo Prob:

P(flipping heads 4 times in a row)?

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Experimental Probability

= What actually happens when we do the experiment.

​

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Law of Large Numbers

​

​

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Law of Large Numbers

​

​

If you roll 1 die, what is the average roll?

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Law of Large Numbers

​

​

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Expected Value

= An estimate of the amount that will be gained/loss in a game of chance.

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Expected Value

= It is calculated by multiplying the probability of winning by the number of points won each time.

​

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Expected Value

= It is the amount you expect to win or lose when playing a game of probability.

​

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Expected Value

= It is the average over a bunch of times.

​

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Expected Value

= What would the EV be if the game is fair?

​

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Expected Value

= What would the EV be if you would lose at the game? Win at the game?

→ Calculate who the game favors

​

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Darts

What is the radius of each circle?

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Darts

What is the radius of each circle?

Center

Radius

5

​

4

​

3

​

2

​

1

​

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Darts

What is the radius of each circle?

Center

Radius

5

1

4

5

3

9

2

13

1

17

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Darts

What is the area of each part?

Center

Radius

Area

5

1

​

4

5

​

3

9

​

2

13

​

1

17

​

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Darts

What is the area of each part?

Center

Radius

Area

5

1

1/289

4

5

24/289

3

9

56/289

2

13

88/289

1

17

120/289

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Darts - Let’s make a table

Event

5

4

3

2

1

Prize (x)

​

​

​

​

​

Prob P(x)

​

​

​

​

​

Expected Value

​

​

​

​

​

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Darts

Event

5

4

3

2

1

Prize (x)

+5

+4

+3

+2

+1

Prob P(x)

1/289

24/289

56/289

88/289

120/289

Expected Value

5(1/289)

4(24/289)

3(56/289)

2(88/289)

1(120/289)

Sum of EV = 1.96

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Darts - An Experiment

50 Trials: Find the average value of the experiment.

How does this compare to EV?

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Practice Time

  • Finish # 2-6
  • Hmwk: p. 943 #13, 17, 18, 22, 36