Summaries

Session 1.3

26^th August 2009

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

A Fair, Three-sided Die

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

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

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”.

1. Additive Rule – Map Faces to Faces

2. Inheritance of Fair Model

3. Fair d3 Model from Fair d6 Model

In the samples, compare the sample proportions (p) to the model probabilities (P) listed above.

Samples 6.30

Sample #1						Sample #2
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	8	0.16				1	6	0.12
2	9	0.18	1	17	0.34	2	6	0.12	1	12	0.24
3	9	0.18				3	12	0.24
4	8	0.16	2	17	0.34	4	8	0.16	2	20	0.4
5	10	0.2				5	6	0.12
6	6	0.12	3	16	0.32	6	12	0.24	3	18	0.36
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Sample #3						Sample #4
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	11	0.22				1	4	0.08
2	13	0.26	1	24	0.48	2	4	0.08	1	8	0.16
3	11	0.22				3	9	0.18
4	3	0.06	2	14	0.28	4	6	0.12	2	15	0.3
5	7	0.14				5	11	0.22
6	5	0.1	3	12	0.24	6	16	0.32	3	27	0.54
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Sample #5						Sample #6
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	5	0.1				1	8	0.16
2	11	0.22	1	16	0.32	2	9	0.18	1	17	0.34
3	13	0.26				3	9	0.18
4	4	0.08	2	17	0.34	4	8	0.16	2	17	0.34
5	8	0.16				5	9	0.18
6	9	0.18	3	17	0.34	6	7	0.14	3	16	0.32
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Pooled135						Pooled246
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	24	0.16				1	18	0.12
2	33	0.22	1	57	0.38	2	19	0.127	1	37	0.247
3	33	0.22				3	30	0.2
4	15	0.1	2	48	0.32	4	22	0.147	2	52	0.347
5	25	0.167				5	26	0.173
6	20	0.133	3	45	0.3	6	35	0.233	3	61	0.407
Total	150	1	Total	150	1	Total	150	1	Total	150	1

Pooled12
FV6	n6	p6	FV3	n3	p3
1	14	0.14
2	15	0.15	1	29	0.29
3	21	0.21
4	16	0.16	2	37	0.37
5	16	0.16
6	18	0.18	3	34	0.34
Total	100	1	Total	100	1
Pooled34
FV6	n6	p6	FV3	n3	p3
1	15	0.15
2	17	0.17	1	32	0.32
3	20	0.2
4	9	0.09	2	29	0.29
5	18	0.18
6	21	0.21	3	39	0.39
Total	100	1	Total	100	1
Pooled56
FV6	n6	p6	FV3	n3	p3
1	13	0.13
2	20	0.2	1	33	0.33
3	22	0.22
4	12	0.12	2	34	0.34
5	17	0.17
6	16	0.16	3	33	0.33
Total	100	1	Total	100	1
Pooled123456
FV6	n6	p6	FV3	n3	p3
1	42	0.14
2	52	0.173	1	94	0.313
3	63	0.21
4	37	0.123	2	100	0.333
5	51	0.17
6	55	0.183	3	106	0.353
Total	300	1	Total	300	1

Samples 8.00

Sample #1						Sample #2
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	6	0.12				1	10	0.2
2	9	0.18	1	15	0.3	2	8	0.16	1	18	0.36
3	15	0.3				3	8	0.16
4	8	0.16	2	23	0.46	4	4	0.08	2	12	0.24
5	2	0.04				5	8	0.16
6	10	0.2	3	12	0.24	6	12	0.24	3	20	0.4
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Sample #3						Sample #4
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	8	0.16				1	4	0.08
2	9	0.18	1	17	0.34	2	8	0.16	1	12	0.24
3	8	0.16				3	7	0.14
4	5	0.1	2	13	0.26	4	13	0.26	2	20	0.4
5	7	0.14				5	10	0.2
6	13	0.26	3	20	0.4	6	8	0.16	3	18	0.36
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Sample #5						Sample #6
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	7	0.14				1	15	0.3
2	7	0.14	1	14	0.28	2	8	0.16	1	23	0.46
3	9	0.18				3	11	0.22
4	9	0.18	2	18	0.36	4	5	0.1	2	16	0.32
5	9	0.18				5	5	0.1
6	9	0.18	3	18	0.36	6	6	0.12	3	11	0.22
Total	50	1	Total	50	1	Total	50	1	Total	50	1
Pooled135						Pooled246
FV6	n6	p6	FV3	n3	p3	FV6	n6	p6	FV3	n3	p3
1	21	0.14				1	29	0.193
2	25	0.167	1	46	0.31	2	24	0.16	1	53	0.353
3	32	0.213				3	26	0.173
4	22	0.147	2	54	0.36	4	22	0.147	2	48	0.32
5	18	0.12				5	23	0.153
6	32	0.213	3	50	0.33	6	26	0.173	3	49	0.327
Total	150	1	Total	150	1	Total	150	1	Total	150	1

Pooled12
FV6	n6	p6	FV3	n3	p3
1	16	0.16
2	17	0.17	1	33	0.33
3	23	0.23
4	12	0.12	2	35	0.35
5	10	0.1
6	22	0.22	3	32	0.32
Total	100	1	Total	100	1
Pooled34
FV6	n6	p6	FV3	n3	p3
1	12	0.12
2	17	0.17	1	29	0.29
3	15	0.15
4	18	0.18	2	33	0.33
5	17	0.17
6	21	0.21	3	38	0.38
Total	100	1	Total	100	1
Pooled56
FV6	n6	p6	FV3	n3	p3
1	22	0.22
2	15	0.15	1	37	0.37
3	20	0.2
4	14	0.14	2	34	0.34
5	14	0.14
6	15	0.15	3	29	0.29
Total	100	1	Total	100	1
Pooled123456
FV6	n6	p6	FV3	n3	p3
1	50	0.167
2	49	0.163	1	99	0.33
3	58	0.193
4	44	0.147	2	102	0.34
5	41	0.137
6	58	0.193	3	99	0.33
Total	300	1	Total	300	1

Pooled Across Sessions (6.30 + 8.00)

Pooled135
FV6	n6	p6	FV3	n3	p3
1	45	0.15(versus .1667)
2	58	0.193(versus .1667)	1	103	0.34(versus .3333)
3	65	0.217(versus .1667)
4	37	0.123(versus .1667)	2	102	0.34(versus .3333)
5	43	0.143(versus .1667)
6	52	0.173(versus .1667)	3	95	0.32(versus .3333)
Total	300	1	Total	300	1

Poole 246
FV6	n6	p6	FV3	n3	p3
1	47	0.157(versus .1667)
2	43	0.143(versus .1667)	1	90	0.3(versus .3333)
3	56	0.187(versus .1667)
4	44	0.147(versus .1667)	2	100	0.333(versus .3333)
5	49	0.163(versus .1667)
6	61	0.203(versus .1667)	3	110	0.367(versus .3333)
Total	300	1	Total	300	1

Pooled All
FV6	n6	p6	FV3	n3	p3
1	92	0.153(versus .1667)
2	101	0.168(versus .1667)	1	193	0.322(versus .3333)
3	121	0.202(versus .1667)
4	81	0.135(versus .1667)	2	202	0.337(versus .3333)
5	92	0.153(versus .1667)
6	113	0.188(versus .1667)	3	205	0.342(versus .3333)
Total	600	1	Total	600	1