Markey's Casino

0

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2

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5

Ace 2 3 4 5 6 7 8 9 10 J Q K

Card

Fre

quen

cy

0

1

2

3

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5

Ace 2 3 4 5 6 7 8 9 10 J Q K

Card

Fre

quen

cyWhat is the probability of picking an ace?

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Ace 2 3 4 5 6 7 8 9 10 J Q K

Card

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cy

Probability =

0

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Ace 2 3 4 5 6 7 8 9 10 J Q K

Card

Fre

quen

cyWhat is the probability of picking an ace?

4 / 52 = .077 or 7.7 chances in 100

0

1

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Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cyEvery card has the same probability of being picked

0

1

2

3

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Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cyWhat is the probability of getting a 10, J, Q, or K?

0

1

2

3

4

5

Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cy(.077) + (.077) + (.077) + (.077) = .308

16 / 52 = .308

0

1

2

3

4

5

Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cyWhat is the probability of getting a 2 and then after replacing the card getting a 3 ?

0

1

2

3

4

5

Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cy(.077) * (.077) = .0059

0

1

2

3

4

5

Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cyWhat is the probability that the two cards you draw will be a black jack?

0

1

2

3

4

5

Ace

(.0

77)

2 (.

077)

3 (.

077)

4 (.

077)

5 (.

077)

6 (.

077)

7 (.

077)

8 (.

077)

9 (.

077)

10 (

.077

)

J (.

077)

Q (

.077

)

K (

.077

)

Card

Fre

quen

cy10 Card = (.077) + (.077) + (.077) + (.077) = .308

Ace after one card is removed = 4/51 = .078

(.308)*(.078) = .024

Practice

• What is the probability of rolling a “1” using a six sided dice?

• What is the probability of rolling either a “1” or a “2” with a six sided dice?

• What is the probability of rolling two “1’s” using two six sided dice?

Practice

• What is the probability of rolling a “1” using a six sided dice?1 / 6 = .166

• What is the probability of rolling either a “1” or a “2” with a six sided dice?


Practice


• What is the probability of rolling either a “1” or a “2” with a six sided dice?(.166) + (.166) = .332


Practice


• What is the probability of rolling either a “1” or a “2” with a six sided dice?(.166) + (.166) = .332

• What is the probability of rolling two “1’s” using two six sided dice?(.166)(.166) = .028

Next step

• Is it possible to apply probabilities to a normal distribution?

Theoretical Normal Curve

-3 -2 -1 1 2 3


-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

We can use the theoretical normal distribution to determine the probability of an event. For example, do you know the probability of getting a Z score of 0 or less?

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.50

We can use the theoretical normal distribution to determine the probability of an event. For example, you know the probability of getting a Z score of 0 or less.

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.50

With the theoretical normal distribution we know the probabilities associated with every z score! The probability of getting a score between a 0 and a 1 is

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

What is the probability of getting a score of 1 or higher?

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

These values are given in Table C on page 384

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

To use this table look for the Z score in column AColumn B is the area between that score and the mean

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

Column B

To use this table look for the Z score in column AColumn C is the area beyond the Z score

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

Column C

The curve is symmetrical -- so the answer for a positive Z score is the same for a negative Z score

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

Column C

Column B

Practice

• What proportion of the normal distribution is found in the following areas (hint: draw out the answer)?

• Between mean and z = .56?

• Beyond z = 2.25?

• Between the mean and z = -1.45

Practice


• Between mean and z = .56?.2123

• Beyond z = 2.25?


Practice



• Beyond z = 2.25?.0122


Practice



• Beyond z = 2.25?.0122

• Between the mean and z = -1.45.4265

Practice

• What proportion of this class would have received an A on the last test if I gave A’s to anyone with a z score of 1.25 or higher?

• .1056

Note• This is using a hypothetical distribution

• Due to chance, empirical distributions are not always identical to theoretical distributions

• If you sampled an infinite number of times they would be equal!

• The theoretical curve represents the “best estimate” of The theoretical curve represents the “best estimate” of how the events would actually occurhow the events would actually occur

0

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4

5

Ace 2 3 4 5 6 7 8 9 10 J Q K

Card

Fre

quen

cyTheoretical Distribution

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Ace 2 3 4 5 6 7 8 9 10 J Q K

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cyEmpirical Distribution based on 52 draws

Empirical Distribution

BFISUR

4.88

4.63

4.38

4.13

3.88

3.63

3.38

3.13

2.88

2.63

2.38

2.13

1.88

1.63

1.38

1.13

Co

un

t50

40

30

20

10

0

BFIOPN

5.00

4.80

4.60

4.40

4.20

4.00

3.80

3.60

3.40

3.20

3.00

2.80

2.60

2.40

2.20

2.00

1.60

Co

un

t

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30

20

10

0


BFISTB

4.88

4.50

4.25

4.00

3.75

3.50

3.25

3.00

2.75

2.50

2.25

2.00

1.75

1.50

1.25

Co

un

t40

30

20

10

0


PROGRAM

http://www.jcu.edu/math/isep/Quincunx/Quincunx.html


Normality frequently occurs in many situations of psychology, and other sciences

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