Summaries

Session 1.3

31^st January 2011

Continue 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 Tables

6:30 Samples

	Sample #1							Sample #2				Pooled 12
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	10	0.2000	0.1667					6	0.1200			16	0.1600
2	5	0.1000	0.1667	1	15	0.3000	0.3333	9	0.1800	15	0.3000	14	0.1400	30	0.3000
3	15	0.3000	0.1667					7	0.1400			22	0.2200
4	8	0.1600	0.1667	2	23	0.4600	0.3333	9	0.1800	16	0.3200	17	0.1700	39	0.3900
5	5	0.1000	0.1667					11	0.2200			16	0.1600
6	7	0.1400	0.1667	3	12	0.2400	0.3333	8	0.1600	19	0.3800	15	0.1500	31	0.3100
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Sample #3							Sample #4				Pooled 34
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	7	0.1400	0.1667					11	0.2200			18	0.1800
2	10	0.2000	0.1667	1	17	0.3400	0.3333	11	0.2200	22	0.4400	21	0.2100	39	0.3900
3	3	0.0600	0.1667					5	0.1000			8	0.0800
4	10	0.2000	0.1667	2	13	0.2600	0.3333	8	0.1600	13	0.2600	18	0.1800	26	0.2600
5	8	0.1600	0.1667					12	0.2400			20	0.2000
6	12	0.2400	0.1667	3	20	0.4000	0.3333	3	0.0600	15	0.3000	15	0.1500	35	0.3500
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Sample #5							Sample #6				Pooled 56
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	10	0.2000	0.1667					12	0.2400			22	0.2200
2	7	0.1400	0.1667	1	17	0.3400	0.3333	6	0.1200	18	0.3600	13	0.1300	35	0.3500
3	7	0.1400	0.1667					7	0.1400			14	0.1400
4	8	0.1600	0.1667	2	15	0.3000	0.3333	8	0.1600	15	0.3000	16	0.1600	30	0.3000
5	10	0.2000	0.1667					9	0.1800			19	0.1900
6	8	0.1600	0.1667	3	18	0.3600	0.3333	8	0.1600	17	0.3400	16	0.1600	35	0.3500
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Pooled 135							Pooled 246				Pooled All
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	27	0.1800	0.1667					29	0.1933			56	0.1867
2	22	0.1467	0.1667	1	49	0.3267	0.3333	26	0.1733	55	0.3667	48	0.1600	104	0.3467
3	25	0.1667	0.1667					19	0.1267			44	0.1467
4	26	0.1733	0.1667	2	51	0.3400	0.3333	25	0.1667	44	0.2933	51	0.1700	95	0.3167
5	23	0.1533	0.1667					32	0.2133			55	0.1833
6	27	0.1800	0.1667	3	50	0.3333	0.3333	19	0.1267	51	0.3400	46	0.1533	101	0.3367
Total	150	1.0000			150	1.0000		150	1.0000	150	1.0000	300	1.0000	300	1.0000

8:00 Samples

	Sample #1							Sample #2				Pooled 12
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	10	0.2000	0.1667					10	0.2000			20	0.2000
2	10	0.2000	0.1667	1	20	0.4000	0.3333	6	0.1200	16	0.3200	16	0.1600	36	0.3600
3	4	0.0800	0.1667					9	0.1800			13	0.1300
4	7	0.1400	0.1667	2	11	0.2200	0.3333	12	0.2400	21	0.4200	19	0.1900	32	0.3200
5	12	0.2400	0.1667					8	0.1600			20	0.2000
6	7	0.1400	0.1667	3	19	0.3800	0.3333	5	0.1000	13	0.2600	12	0.1200	32	0.3200
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Sample #3							Sample #4				Pooled 34
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	5	0.1000	0.1667					6	0.1200			11	0.1100
2	7	0.1400	0.1667	1	12	0.2400	0.3333	10	0.2000	16	0.3200	17	0.1700	28	0.2800
3	5	0.1000	0.1667					11	0.2200			16	0.1600
4	10	0.2000	0.1667	2	15	0.3000	0.3333	9	0.1800	20	0.4000	19	0.1900	35	0.3500
5	11	0.2200	0.1667					7	0.1400			18	0.1800
6	12	0.2400	0.1667	3	23	0.4600	0.3333	7	0.1400	14	0.2800	19	0.1900	37	0.3700
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Sample #5							Sample #6				Pooled 56
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	6	0.1200	0.1667					7	0.1400			13	0.1300
2	8	0.1600	0.1667	1	14	0.2800	0.3333	9	0.1800	16	0.3200	17	0.1700	30	0.3000
3	16	0.3200	0.1667					10	0.2000			26	0.2600
4	9	0.1800	0.1667	2	25	0.5000	0.3333	4	0.0800	14	0.2800	13	0.1300	39	0.3900
5	4	0.0800	0.1667					13	0.2600			17	0.1700
6	7	0.1400	0.1667	3	11	0.2200	0.3333	7	0.1400	20	0.4000	14	0.1400	31	0.3100
Total	50	1.0000			50	1.0000		50	1.0000	50	1.0000	100	1.0000	100	1.0000
	Pooled 135							Pooled 246				Pooled All
d6	n	p	P	d3	n	p	P	n	p	n	p	n	p	n	p
1	21	0.1400	0.1667					23	0.1533			44	0.1467
2	25	0.1667	0.1667	1	46	0.3067	0.3333	25	0.1667	48	0.3200	50	0.1667	94	0.3133
3	25	0.1667	0.1667					30	0.2000			55	0.1833
4	26	0.1733	0.1667	2	51	0.3400	0.3333	25	0.1667	55	0.3667	51	0.1700	106	0.3533
5	27	0.1800	0.1667					28	0.1867			55	0.1833
6	26	0.1733	0.1667	3	53	0.3533	0.3333	19	0.1267	47	0.3133	45	0.1500	100	0.3333
Total	150	1.0000			150	1.0000		150	1.0000	150	1.0000	300	1.0000	300	1.0000

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