Summaries

Session 1.3

2^nd June 2010

Begin work on the Long Run Argument and Perfect Sample case types in 1^st Hourly Stuff. Start to build your narratives.

We extend our study of probability to dice. We revisit the idea of a model or population proportion as a probability, and introduce the idea of a random variable.

Models

A Fair, Six-sided Die

Face Value, d6 (FV d6)	Probability
1	1/6
2	1/6
3	1/6
4	1/6
5	1/6
6	1/6

Using a Fair, Six-sided Die to Simulate A Fair, Three-sided Die

Face Value, d6 (FV d6)	Mapped Face Value, d3 (FV d3)
1	1
2
3	2
4
5	3
6

A Fair, Three-sided Die

Face Value, d3 (FV d3)	Probability
1	1/3
2	1/3
3	1/3

Probability Calculations (fair d6→ fair d3)

Pr{E} denotes Probability for the event E.

The Fair d6 Model

FV: Face Values: 1,2,3,4,5,6

Fair Model: Equally likely face values – 1/6 per face value

Pr{d6 Shows 1} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “1”.	Pr{d6 Shows 2} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “2”.	Pr{d6 Shows 3} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “3”.
Pr{d6 Shows 4} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “4”.	Pr{d6 Shows 5} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “5”.	Pr{d6 Shows 6} = (1/6) @ .1667 or 16.67% In long runs of tosses, approximately 1 toss in 6 shows “6”.

The Fair d3 Model Nested within a Fair d6 Model

FV: Face Values: 1(1,2), 2(3,4), 3(5,6)

Fair Model: Equally likely face values –  (2/6 =)1/3 per face value.

Pr{d3 shows “1”} = Pr{d6 Shows 1} + Pr{d6 Shows 2}1 = (1/6) + (1/6) = 2/6 = 1/3 @ .3333 or 33.33% In long runs of tosses, approximately 1 toss in 3 shows “1”.

Pr{d3 shows “2”} = Pr{d6 Shows 3} + Pr{d6 Shows 4} = (1/6) + (1/6)2 = 2/6 = 1/3 @ .3333 or 33.33% In long runs of tosses, approximately 1 toss in 3 shows “2”.

Pr{d3 shows “3”} = Pr{d6 Shows 5} + Pr{d6 Shows 6} = (1/6) + (1/6) = 2/6 = 1/33 @ .3333 or 33.33% In long runs of tosses, approximately 1 toss in 3 shows “3”.

Probability Computational Rules

1. Additive Rule – Map Faces to Faces

2. Inheritance of Fair Model

3. Fair d3 Model from Fair d6 Model

D6/D3 Worksheet

50 Tosses per Sample (n=50)

Sample Grid – One Toss per Cell

0X	X2	X3	X4	X5	X6	X7	X8	X9	X9
2									X
3									X
4									X
5

Case Steps:

Toss Die

Note D6 Face Value

Map D6 to D3 and Note D3 Face Value:

D6 Face Value Þ D3 Face Value

1, 2 Þ 1

3, 4 Þ 2

5, 6 Þ 3

D6 Face Value	Count	D3 Face Value	Count
1		1
2
3		2
4
5		3
6
Total		Total

Sample Table

	Sample #1					Sample #2				Pooled 12
d6	n	p	d3	n	p	n	p	n	p	n	p	n	p
1	9	9/50=0.18				7	0.14			9+7=16	0.16
2	10	10/50=0.20	1	9+10=19	0.38	8	0.16	15	0.30	10+8=18	0.18	34	34/100=0.34
3	10	10/50=0.20				9	0.18			10+9=19	0.19
4	5	5/50=0.10	2	10+5=15	0.30	6	0.12	15	0.30	5+6=11	0.11	30	30/100=0.30
5	7	7/50=0.14				10	0.20			7+10=17	0.17
6	9	9/18=0.18	3	7+9=16	0.32	10	0.20	20	0.40	9+10=19	0.19	36	36/100=0.36
Total	50	50/50=1.00		50	1.00	50	1.00	50	1.00	50+50=100	1.00	100	1.00
	Sample #3					Sample #4				Pooled 34
d6	n	p	d3	n	p	n	p	n	p	n	p	n	P
1	12	0.24				12	0.24			24	0.24
2	8	0.16	1	20	0.40	6	0.12	18	0.36	14	0.14	38	0.38
3	11	0.22				9	0.18			20	0.20
4	6	0.12	2	17	0.34	7	0.14	16	0.32	13	0.13	33	0.33
5	8	0.16				8	0.16			16	0.16
6	5	0.10	3	13	0.26	8	0.16	16	0.32	13	0.13	29	0.29
Total	50	1.00		50	1.00	50	1.00	50	1.00	100	1.00	100	1.00
	Sample #5					Sample #6				Pooled 56
d6	n	p	d3	n	p	n	p	n	p	n	p	n	P
1	8	0.16				7	0.14			15	0.15
2	11	0.22	1	19	0.38	10	0.20	17	0.34	21	0.21	36	0.36
3	8	0.16				9	0.18			17	0.17
4	10	0.20	2	18	0.36	5	0.10	14	0.28	15	0.15	32	0.32
5	6	0.12				9	0.18			15	0.15
6	7	0.14	3	13	0.26	10	0.20	19	0.38	17	0.17	32	0.32
Total	50	1.00		50	1.00	50	1.00	50	1.00	100	1.00	100	1.00
	Pooled 135					Pooled 246				Pooled All
d6	n	p	d3	n	p	n	p	n	p	n	p	n	P
1	29	0.19				26	0.17			55	0.18
2	29	0.19	1	58	0.39	24	0.16	50	0.33	53	0.18	55+53=108	108/300=0.36
3	29	0.19				27	0.18			56	0.19
4	21	0.14	2	50	0.33	18	0.12	45	0.30	39	0.13	56+39=95	95/300=0.32
5	21	0.14				27	0.18			48	0.16
6	21	0.14	3	42	0.28	28	0.19	55	0.37	49	0.16	48+49=97	97/300=0.32
Total	150	1.00		150	1.00	150	1.00	150	1.00	300	1.00	300	1.00

Compare the correspondence of the sample proportions (p) to the model probabilities (P).

Fair Models
d6	N	P	d3	N	P
1	1	1/6≈0.1667
2	1	1/6≈0.1667	1	1	1/3≈0.3333
3	1	1/6≈0.1667
4	1	1/6≈0.1667	2	1	1/3≈0.3333
5	1	1/6≈0.1667
6	1	1/6≈0.1667	3	1	1/3≈0.3333
Total	6	6/6=1.0000		3	3/3=1.0000