Summaries

Session 1.3

25^th August 2010

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	7	0.1400	0.1667					7	0.1400			14	0.1400
2	7	0.1400	0.1667	1	14	0.2800	0.3333	15	0.3000	22	0.4400	22	0.2200	36	0.3600
3	8	0.1600	0.1667					3	0.0600			11	0.1100
4	9	0.1800	0.1667	2	17	0.3400	0.3333	14	0.2800	17	0.3400	23	0.2300	34	0.3400
5	13	0.2600	0.1667					7	0.1400			20	0.2000
6	6	0.1200	0.1667	3	19	0.3800	0.3333	4	0.0800	11	0.2200	10	0.1000	30	0.3000
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	10	0.2000	0.1667					7	0.1400			17	0.1700
2	5	0.1000	0.1667	1	15	0.3000	0.3333	13	0.2600	20	0.4000	18	0.1800	35	0.3500
3	7	0.1400	0.1667					5	0.1000			12	0.1200
4	10	0.2000	0.1667	2	17	0.3400	0.3333	5	0.1000	10	0.2000	15	0.1500	27	0.2700
5	6	0.1200	0.1667					10	0.2000			16	0.1600
6	12	0.2400	0.1667	3	18	0.3600	0.3333	10	0.2000	20	0.4000	22	0.2200	38	0.3800
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					4	0.0800			10	0.1000
2	14	0.2800	0.1667	1	20	0.4000	0.3333	8	0.1600	12	0.2400	22	0.2200	32	0.3200
3	6	0.1200	0.1667					6	0.1200			12	0.1200
4	9	0.1800	0.1667	2	15	0.3000	0.3333	12	0.2400	18	0.3600	21	0.2100	33	0.3300
5	9	0.1800	0.1667					12	0.2400			21	0.2100
6	6	0.1200	0.1667	3	15	0.3000	0.3333	8	0.1600	20	0.4000	14	0.1400	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	23	0.1533	0.1667					18	0.1200			41	0.1367
2	26	0.1733	0.1667	1	49	0.3267	0.3333	36	0.2400	54	0.3600	62	0.2067	103	0.3433
3	21	0.1400	0.1667					14	0.0933			35	0.1167
4	28	0.1867	0.1667	2	49	0.3267	0.3333	31	0.2067	45	0.3000	59	0.1967	94	0.3133
5	28	0.1867	0.1667					29	0.1933			57	0.1900
6	24	0.1600	0.1667	3	52	0.3467	0.3333	22	0.1467	51	0.3400	46	0.1533	103	0.3433
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	11	0.2200	0.1667					7	0.1400			18	0.1800
2	9	0.1800	0.1667	1	20	0.4000	0.3333	9	0.1800	16	0.3200	18	0.1800	36	0.3600
3	11	0.2200	0.1667					6	0.1200			17	0.1700
4	5	0.1000	0.1667	2	16	0.3200	0.3333	10	0.2000	16	0.3200	15	0.1500	32	0.3200
5	11	0.2200	0.1667					8	0.1600			19	0.1900
6	3	0.0600	0.1667	3	14	0.2800	0.3333	10	0.2000	18	0.3600	13	0.1300	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	9	0.1800	0.1667					8	0.1600			17	0.1700
2	7	0.1400	0.1667	1	16	0.3200	0.3333	9	0.1800	17	0.3400	16	0.1600	33	0.3300
3	7	0.1400	0.1667					11	0.2200			18	0.1800
4	8	0.1600	0.1667	2	15	0.3000	0.3333	9	0.1800	20	0.4000	17	0.1700	35	0.3500
5	8	0.1600	0.1667					7	0.1400			15	0.1500
6	11	0.2200	0.1667	3	19	0.3800	0.3333	6	0.1200	13	0.2600	17	0.1700	32	0.3200
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	8	0.1600	0.1667					7	0.1400			15	0.1500
2	13	0.2600	0.1667	1	21	0.4200	0.3333	7	0.1400	14	0.2800	20	0.2000	35	0.3500
3	7	0.1400	0.1667					11	0.2200			18	0.1800
4	5	0.1000	0.1667	2	12	0.2400	0.3333	6	0.1200	17	0.3400	11	0.1100	29	0.2900
5	10	0.2000	0.1667					8	0.1600			18	0.1800
6	7	0.1400	0.1667	3	17	0.3400	0.3333	11	0.2200	19	0.3800	18	0.1800	36	0.3600
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	28	0.1867	0.1667					22	0.1467			50	0.1667
2	29	0.1933	0.1667	1	57	0.3800	0.3333	25	0.1667	47	0.3133	54	0.1800	104	0.3467
3	25	0.1667	0.1667					28	0.1867			53	0.1767
4	18	0.1200	0.1667	2	43	0.2867	0.3333	25	0.1667	53	0.3533	43	0.1433	96	0.3200
5	29	0.1933	0.1667					23	0.1533			52	0.1733
6	21	0.1400	0.1667	3	50	0.3333	0.3333	27	0.1800	50	0.3333	48	0.1600	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