Summaries

Session 1.3

25^th January 2010

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

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

Samples 6.30

n=50

Sample #1						Sample #2
d6	n	p	d3	n	p	d6	n	p	d3	n	p
1	7	0.14				1	10	0.20
2	8	0.16	1	15	0.30	2	5	0.10	1	15	0.30
3	11	0.22				3	8	0.16
4	8	0.16	2	19	0.38	4	9	0.18	2	17	0.34
5	10	0.20				5	9	0.18
6	6	0.12	3	16	0.32	6	9	0.18	3	18	0.36
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00
Sample #3						Sample #4
d6	n	p	d3	n	p	d6	n	p	d3	n	p
1	5	0.10				1	12	0.24
2	5	0.10	1	10	0.20	2	6	0.12	1	18	0.36
3	5	0.10				3	10	0.20
4	11	0.22	2	16	0.32	4	8	0.16	2	18	0.36
5	14	0.28				5	5	0.10
6	10	0.20	3	24	0.48	6	9	0.18	3	14	0.28
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00
Sample #5						Sample #6
d6	n	p	d3	n	p	d6	n	p	d3	n	p
1	7	0.14				1	4	0.08
2	10	0.20	1	17	0.34	2	13	0.26	1	17	0.34
3	10	0.20				3	14	0.28
4	4	0.08	2	14	0.28	4	3	0.06	2	17	0.34
5	7	0.14				5	7	0.14
6	12	0.24	3	19	0.38	6	9	0.18	3	16	0.32
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00

n=100

Pooled 12
d6	n	p	d3	n	p
1	17	0.17
2	13	0.13	1	30	0.30
3	19	0.19
4	17	0.17	2	36	0.36
5	19	0.19
6	15	0.15	3	34	0.34
Total	100	1.00		100	1.00
Pooled 34
d6	n	p	d3	n	p
1	17	0.17
2	11	0.11	1	28	0.28
3	15	0.15
4	19	0.19	2	34	0.34
5	19	0.19
6	19	0.19	3	38	0.38
Total	100	1.00		100	1.00
Pooled 56
d6	n	p	d3	n	P
1	11	0.11
2	23	0.23	1	34	0.34
3	24	0.24
4	7	0.07	2	31	0.31
5	14	0.14
6	21	0.21	3	35	0.35
Total	100	1.00		100	1.00

n=150

Pooled 135						Pooled 246
d6	n	P	d3	n	p	d6	n	p	d3	n	p
1	19	19/150 ≈0.13				1	26	26/150 ≈0.17
2	23	23/150 ≈0.15	1	42	0.28	2	24	24/150 ≈0.16	1	50	0.33
3	26	26/150 ≈0.17				3	32	32/150 ≈0.21
4	23	23/150 ≈0.15	2	49	0.33	4	20	20/150 ≈0.13	2	52	0.35
5	31	31/150 ≈0.21				5	21	21/150 ≈0.14
6	28	28/150 ≈0.19	3	59	0.39	6	27	27/150 ≈0.18	3	48	0.32
Total	150	1.00		150	1.00	Total	150	1.00		150	1.00

n=300

Pooled All
d6	n	p	d3	n	p
1	45	45/300≈0.15
2	47	47/300≈0.16	1	92	92/300≈0.31
3	58	58/300≈0.19
4	43	43/300≈0.14	2	101	101/300≈0.34
5	52	52/300≈0.17
6	55	55/300≈0.18	3	107	107/300≈0.36
Total	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	1	1	1/3 ≈ 0.3333
2	1	1/6 ≈ 0.1667
3	1	1/6 ≈ 0.1667	2	1	1/3 ≈ 0.3333
4	1	1/6 ≈ 0.1667
5	1	1/6 ≈ 0.1667	3	1	1/3 ≈ 0.3333
6	1	1/6 ≈ 0.1667
Total	6	1.0000		3	1.0000

Samples 8:00

n=50

Sample #1						Sample #2
d6	N	p	d3	n	p	d6	n	p	d3	n	p
1	5	5/50 = 0.10				1	10	0.20
2	10	10/50 = 0.20	1	15	0.30	2	7	0.14	1	17	17/50 = 0.34
3	11	11/50 = 0.22				3	9	0.18
4	11	11/50 = 0.22	2	22	0.44	4	9	0.18	2	18	18/50 = 0.36
5	8	8/50 = 0.16				5	9	0.18
6	5	5/50 = 0.10	3	13	0.26	6	6	0.12	3	15	15/50 = 0.30
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00
Sample #3						Sample #4
d6	N	p	d3	n	p	d6	n	p	d3	n	p
1	6	0.12				1	10	0.20
2	12	0.24	1	18	0.36	2	8	0.16	1	18	0.36
3	9	0.18				3	10	0.20
4	8	0.16	2	17	0.34	4	7	0.14	2	17	0.34
5	7	0.14				5	7	0.14
6	8	0.16	3	15	0.30	6	8	0.16	3	15	0.30
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00
Sample #5						Sample #6
d6	N	p	d3	n	p	d6	n	p	d3	n	p
1	6	0.12				1	8	0.16
2	5	0.10	1	11	0.22	2	9	0.18	1	17	0.34
3	8	0.16				3	7	0.14
4	11	0.22	2	19	0.38	4	11	0.22	2	18	0.36
5	8	0.16				5	10	0.20
6	12	0.24	3	20	0.40	6	5	0.10	3	15	0.30
Total	50	1.00		50	1.00	Total	50	1.00		50	1.00

n=100

Pooled 12
d6	n	p	d3	n	P
1	15	15/100=0.15
2	17	17/100=0.17	1	32	0.32
3	20	20/100=0.20
4	20	20/100=0.20	2	40	0.40
5	17	17/100=0.17
6	11	11/100=0.11	3	28	0.28
Total	100	1.00		100	1.00
Pooled 34
d6	n	p	d3	n	P
1	16	0.16
2	20	0.20	1	36	36/100=0.36
3	19	0.19
4	15	0.15	2	34	34/100=0.34
5	14	0.14
6	16	0.16	3	30	30/100=0.30
Total	100	1.00		100	1.00
Pooled 56
d6	n	p	d3	n	p
1	14	0.14
2	14	0.14	1	28	0.28
3	15	0.15
4	22	0.22	2	37	0.37
5	18	0.18
6	17	0.17	3	35	0.35
Total	100	1.00		100	1.00

n=150

Pooled 135						Pooled 246
d6	n	p	d3	n	p	d6	n	P	d3	n	p
1	17	17/150 ≈ 0.11				1	28	28/150 ≈ 0.19
2	27	27/150 ≈ 0.18	1	44	0.29	2	24	24/150 ≈ 0.16	1	52	0.35
3	28	28/150 ≈ 0.19				3	26	26/150 ≈ 0.17
4	30	30/150 ≈ 0.20	2	58	0.39	4	27	27/150 ≈ 0.18	2	53	0.35
5	23	23/150 ≈ 0.15				5	26	26/150 ≈ 0.17
6	25	25/150 ≈ 0.17	3	48	0.32	6	19	19/150 ≈ 0.13	3	45	0.30
Total	150	1.00		150	1.00	Total	150	1.00		150	1.00

n=300

Pooled All
d6	n	p	d3	n	p
1	45	45/300 ≈0.15
2	51	51/300 ≈0.17	1	96	96/300 ≈0.32
3	54	54/300 ≈0.18
4	57	57/300 ≈0.19	2	111	111/300 ≈0.37
5	49	45/300 ≈0.16
6	44	44/300 ≈0.15	3	93	93/300 ≈0.31
Total	300	1.00		300	1.00

Pooled Across Sections: 6:30 + 8:00

Pooled 135						Pooled 246						Pooled All
d6	n	p	d3	n	p	d6	n	p	d3	n	p	d6	n	p	d3	n	p
1	36	0.12				1	54	0.18				1	90	0.15
2	50	0.17	1	86	0.29	2	48	0.16	1	102	0.34	2	98	0.16	1	188	0.31
3	54	0.18				3	58	0.19				3	112	0.19
4	53	0.18	2	107	0.36	4	47	0.16	2	105	0.35	4	100	0.17	2	212	0.35
5	54	0.18				5	47	0.16				5	101	0.17
6	53	0.18	3	107	0.36	6	46	0.15	3	93	0.31	6	99	0.17	3	200	0.33
Total	300	1.00		300	1.00	Total	300	1.00		300	1.00	Total	600	1.00		600	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	1	1	1/3 ≈ 0.3333
2	1	1/6 ≈ 0.1667
3	1	1/6 ≈ 0.1667	2	1	1/3 ≈ 0.3333
4	1	1/6 ≈ 0.1667
5	1	1/6 ≈ 0.1667	3	1	1/3 ≈ 0.3333
6	1	1/6 ≈ 0.1667
Total	6	1.0000		3	1.0000

Data