Summaries

Session 0.1

18^th August 2010

Session 1.1

Sampling a Simple Population

We use random sampling to estimate an empirical model of a population. We check the empirical model by direct inspection of the population. We repeat sampling with replacement, obtaining multiple random samples from the same population, obtained in the same process. We combine (pool) compatible samples to form larger samples. Pooling samples of size 50, we obtain samples of size 100, 150 and 300. In general, as sample size increases, samples become more precise and reliable, provided that the sampling process is reliable.

Random sampling is the basis for obtaining information in statistical activities. Sampling is necessary, tedious, time consuming and expensive. Random sampling incorporates reliability, precision and uncertainty.

In this session, we begin the study of probability. We begin with a very basic example of a population, and explore the process of sampling a population.

In our first case, we begin with a fair, six-sided die. We track predicted and observed face values in six random samples of 50 tosses of the die. We then compare our samples to what is expected under the fair model.

We examine two modes of sampling a population: census (total enumeration), in which every member of the population is examined; and random sampling with replacement (SRS/WR), in which single members are repeatedly selected from the population. One practical reason why we would want a sampling process is that we wish to estimate some property of the population. Total enumeration allows a definitive settling of the question, and random sampling allows an approximate answer. In most practical settings, the populations of interest are too difficult to totally enumerate – the population is too large, or too complex, or cannot be accessed in total. In practical applications, it is sufficient (and usually necessary) to use a suitable random sample in lieu of the total population.

In our second case, we begin with a color bowl whose true color frequencies are known. We compute a population frequency model and then compute the expected structure for random samples from that bowl. We obtain six (6) random samples, each consisting of 50 draws with replacement (SRS/WR). We then compute sample color frequencies and compare them to the true structure of the bowl.

We then explore a bit of decision theory by playing with Ellsberg’s Urns.

Prediction and Probabilistic Randomness: Predicting the Behavior of a Six-sided Die

Process

We have a fair, six-sided die, with face values 1, 2, 3, 4, 5 and 6. Prior to each toss of the die, a member of the group predicts the face value that will be observed on that toss. Upon tossing the die, the group notes the observed face value, as well as the correctness (or lack thereof) of the prediction. Each group produces a sample of 50 tosses.

Sample Worksheet

Prediction and the Fair Die

Sample Grid n=50

Each cell corresponds to a single toss of the die.

X
X
X
X
X	X	X	X	X	X	X	X	X	X

Sample results are tabulated in the form below.

Face Value	Count	Prediction	Count
1		Hit(Correct)
2		Miss(Incorrrect)
3		Total
4
5
6
Total

Samples – Face Values and Predictions

Here are the results for our six samples. You should be able to begin with the counts in the table and work out the proportions and percentages.

6:30 Samples

Prediction and the Fair Die
Samples				Samples				Pooled
#1				#2				12
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	14	14/50 = 0.28	100*.28=28	1	7	0.14	14	1	14+7=21	21/100=0.21	21
2	6	6/50 = 0.12	100*.12=12	2	10	0.2	20	2	6+10=16	16/100=0.16	16
3	9	9/50 = 0.18	100*.18=18	3	8	0.16	16	3	9+8=17	17/100=0.17	17
4	6	6/50 = 0.12	100*.12=12	4	10	0.2	20	4	6+10=16	16/100=0.16	16
5	9	9/50 = 0.18	100*.18=18	5	5	0.1	10	5	9+5=14	14/100=0.14	14
6	6	6/50 = 0.12	100*.12=12	6	10	0.2	20	6	6+10=16	16/100=0.16	16
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	9	0.18	18	Hit	13	0.26	26	Hit	22	0.22	22
Miss	41	0.82	82	Miss	37	0.74	74	Miss	78	0.78	78
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#3				#4				34
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	4	0.08	8	1	10	0.2	20	1	14	0.14	14
2	12	0.24	24	2	6	0.12	12	2	18	0.18	18
3	7	0.14	14	3	8	0.16	16	3	15	0.15	15
4	6	0.12	12	4	12	0.24	24	4	18	0.18	18
5	14	0.28	28	5	9	0.18	18	5	23	0.23	23
6	7	0.14	14	6	5	0.1	10	6	12	0.12	12
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	8	0.16	16	Hit	6	0.12	12	Hit	14	0.14	14
Miss	42	0.84	84	Miss	44	0.88	88	Miss	86	0.86	86
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#5				#6				56
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	8	0.16	16	1	8	0.16	16	1	16	0.16	16
2	10	0.2	20	2	6	0.12	12	2	16	0.16	16
3	10	0.2	20	3	5	0.1	10	3	15	0.15	15
4	6	0.12	12	4	9	0.18	18	4	15	0.15	15
5	9	0.18	18	5	8	0.16	16	5	17	0.17	17
6	7	0.14	14	6	14	0.28	28	6	21	0.21	21
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	13	0.26	26	Hit	8	0.16	16	Hit	21	0.21	21
Miss	37	0.74	74	Miss	42	0.84	84	Miss	79	0.79	79
Total	50	1	100	Total	50	1	100	Total	100	1	100
Pooled				Pooled				Pooled
135				246				123456
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	26	0.173333333	17.333	1	25	0.166666667	16.667	1	51	51/300 =0.17	17
2	28	0.186666667	18.667	2	22	0.146666667	14.667	2	50	50/300=0.1666667	16.667
3	26	0.173333333	17.333	3	21	0.14	14	3	47	47/300=0.1566667	15.667
4	18	0.12	12	4	31	0.206666667	20.667	4	49	50/300=0.1633333	16.333
5	32	0.213333333	21.333	5	22	0.146666667	14.667	5	54	54/300=0.18	18
6	20	0.133333333	13.333	6	29	0.193333333	19.333	6	49	49/300=0.1633333	16.333
Total	150	1	100	Total	150	1	100	Total	300	1	100
Prediction				Prediction				Prediction
Hit	30	0.2	20	Hit	27	0.18	18	Hit	57	0.19	19
Miss	120	0.8	80	Miss	123	0.82	82	Miss	243	0.81	81
Total	150	1	100	Total	150	1	100	Total	300	1	100

In the fair die model for this case, in long runs of tosses of the die: approximately 16⅔% of tosses show “1”, approximately 16⅔% of tosses show “2”, approximately 16⅔% of tosses show “3”, approximately 16⅔% of tosses show “4”, approximately 16⅔% of tosses show “5”, and approximately 16⅔% of tosses show “6.” The sample data are generally compatible with a fair die assumption (equally-likely face values) and with a baseline expected prediction success rate of (1/6), or 16⅔%. Sample performance seems to improve with increasing sample size – but the samples do not exactly fit the fair assumption.

Sample versus Fair Model

6:30

Face Value 1: 17% (Sample) versus 16.67% (Fair Model)

Face Value 2: 16.67% (Sample) versus 16.67% (Fair Model)

Face Value 3: 15.67% (Sample) versus 16.67% (Fair Model)

Face Value 4: 16.33% (Sample) versus 16.67% (Fair Model)

Face Value 5: 18% (Sample) versus 16.67% (Fair Model)

Face Value 6: 16.33% (Sample) versus 16.67% (Fair Model)

Prediction “Hit”: 19% (Sample) versus 16.67% (Fair Model)

Prediction “Miss”: 81% (Sample) versus 83.33% (Fair Model))

8:00 Samples

Prediction and the Fair Die
Samples				Samples				Pooled
#1				#2				12
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	7	0.14	14	1	6	0.12	12	1	13	0.13	13
2	6	0.12	12	2	5	0.1	10	2	11	0.11	11
3	14	0.28	28	3	10	0.2	20	3	24	0.24	24
4	9	0.18	18	4	7	0.14	14	4	16	0.16	16
5	6	0.12	12	5	16	0.32	32	5	22	0.22	22
6	8	0.16	16	6	6	0.12	12	6	14	0.14	14
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	10	0.2	20	Hit	8	0.16	16	Hit	18	0.18	18
Miss	40	0.8	80	Miss	42	0.84	84	Miss	82	0.82	82
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#3				#4				34
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	6	0.12	12	1	7	0.14	14	1	13	0.13	13
2	5	0.1	10	2	7	0.14	14	2	12	0.12	12
3	12	0.24	24	3	9	0.18	18	3	21	0.21	21
4	8	0.16	16	4	3	0.06	6	4	11	0.11	11
5	5	0.1	10	5	13	0.26	26	5	18	0.18	18
6	14	0.28	28	6	11	0.22	22	6	25	0.25	25
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	8	0.16	16	Hit	6	0.12	12	Hit	14	0.14	14
Miss	42	0.84	84	Miss	44	0.88	88	Miss	86	0.86	86
Total	50	1	100	Total	50	1	100	Total	100	1	100
Samples				Samples				Pooled
#5				#6				56
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	6	0.12	12	1	12	0.24	24	1	18	0.18	18
2	8	0.16	16	2	8	0.16	16	2	16	0.16	16
3	7	0.14	14	3	4	0.08	8	3	11	0.11	11
4	11	0.22	22	4	6	0.12	12	4	17	0.17	17
5	12	0.24	24	5	9	0.18	18	5	21	0.21	21
6	6	0.12	12	6	11	0.22	22	6	17	0.17	17
Total	50	1	100	Total	50	1	100	Total	100	1	100
Prediction				Prediction				Prediction
Hit	7	0.14	14	Hit	9	0.18	18	Hit	16	0.16	16
Miss	43	0.86	86	Miss	41	0.82	82	Miss	84	0.84	84
Total	50	1	100	Total	50	1	100	Total	100	1	100
Pooled				Pooled				Pooled
135				246				123456
Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent	Face Value	Count	Proportion	Percent
1	19	0.126667	12.667	1	25	0.166667	16.667	1	44	0.146667	14.667
2	19	0.126667	12.667	2	20	0.133333	13.333	2	39	0.13	13
3	33	0.22	22	3	23	0.153333	15.333	3	56	0.186667	18.667
4	28	0.186667	18.667	4	16	0.106667	10.667	4	44	0.146667	14.667
5	23	0.153333	15.333	5	38	0.253333	25.333	5	61	0.203333	20.333
6	28	0.186667	18.667	6	28	0.186667	18.667	6	56	0.186667	18.667
Total	150	1	100	Total	150	1	100	Total	300	1	100
Prediction				Prediction				Prediction
Hit	25	0.166667	16.667	Hit	23	0.153333	15.333	Hit	48	0.16	16
Miss	125	0.833333	83.333	Miss	127	0.846667	84.667	Miss	252	0.84	84
Total	150	1	100	Total	150	1	100	Total	300	1	100

In the fair die model for this case, in long runs of tosses of the die: approximately 16⅔% of tosses show “1”, approximately 16⅔% of tosses show “2”, approximately 16⅔% of tosses show “3”, approximately 16⅔% of tosses show “4”, approximately 16⅔% of tosses show “5”, and approximately 16⅔% of tosses show “6.” The sample data are generally compatible with a fair die assumption (equally-likely face values) and with a baseline expected prediction success rate of (1/6), or 16⅔%. Sample performance seems to improve with increasing sample size – but the samples do not exactly fit the fair assumption.

Sample versus Fair Model

8:00

Face Value 1: 14.67% (Sample) versus 16.67% (Fair Model)

Face Value 2: 13% (Sample) versus 16.67% (Fair Model)

Face Value 3: 18.67% (Sample) versus 16.67% (Fair Model)

Face Value 4: 14.67% (Sample) versus 16.67% (Fair Model)

Face Value 5: 20.33% (Sample) versus 16.67% (Fair Model)

Face Value 6: 18.67% (Sample) versus 16.67% (Fair Model)

Prediction “Hit”: 16% (Sample) versus 16.67% (Fair Model)

Prediction “Miss”: 84% (Sample) versus 83.33% (Fair Model))

Case Study 1.1: A Color Bowl

In random sampling, we might get a complete list of colors - we'd need a total sample (census) for that kind of listing. The sample proportions of each listed color approximate the corresponding model proportion in the bowl itself. In census sampling, every object in the bowl is counted. The listing is complete, and the model proportions may be calculated directly.

The basic idea in case study 1.1 is that random samples give imperfect pictures of what is being sampled. However, with sufficiently large samples, these samples can reliably yield good pictures of the processes or populations being sampled. And the essence of many statistical applications is the study of selected processes or populations. For a sense of the efficiency of the samples, compare sample and true percentages.

Process

We have a four color bowl, with blue, green, red and yellow marbles. Prior to each draw from the bowl, the bowl is thoroughly mixed, giving each resident marble an approximately equal chance of selection. After mixing, a blind (made without looking into the bowl) draw of a single marble is made. The group notes the color of the marble, and the marble is returned to the bowl – this is sampling with replacement. The mixing makes the sampling random. Each group produces a sample of 50 tosses.

Each cell corresponds to a single draw with replacement from the bowl.

Sample Grid (n=50)

0

0

0

0

0

0

0

0

0

0

Sample results are tabulated in the form below.

Table – Draws with Replacement

Color	Count
Blue
Green
Red
Yellow
Total

Samples from the Color Bowl

Here are the six samples from our groups. You should be able to begin with the counts in the table and work out the proportions and percentages.

6:30 Samples

Color Bowl I - Sampling with Replacement
#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	20	20/50=0.4	100*.4=40	Blue	25	0.5	50	Blue	20+25=45	45/100=0.45	45	42.9
Green	12	12/50=0.24	100*.24=24	Green	11	0.22	22	Green	12+11=23	23/100=0.23	23	21.4
Red	15	15/50=0.3	100*.3=30	Red	5	0.1	10	Red	15+5=20	20/100=0.2	20	21.4
Yellow	3	3/50=0.06	100*.06=6	Yellow	9	0.18	18	Yellow	3+9=12	12/100=0.12	12	14.3
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#3				#4				Pooled 34
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	27	0.54	54	Blue	21	0.42	42	Blue	48	0.48	48	42.9
Green	7	0.14	14	Green	13	0.26	26	Green	20	0.2	20	21.4
Red	11	0.22	22	Red	9	0.18	18	Red	20	0.2	20	21.4
Yellow	5	0.1	10	Yellow	7	0.14	14	Yellow	12	0.12	12	14.3
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#5				#6				Pooled 56
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	22	0.44	44	Blue	26	0.52	52	Blue	48	0.48	48	42.9
Green	9	0.18	18	Green	11	0.22	22	Green	20	0.2	20	21.4
Red	12	0.24	24	Red	11	0.22	22	Red	23	0.23	23	21.4
Yellow	7	0.14	14	Yellow	2	0.04	4	Yellow	9	0.09	9	14.3
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	69	0.46	46	Blue	72	0.48	48	Blue	141	0.47	47	42.9
Green	28	0.1866667	18.6667	Green	35	0.2333333	23.3333	Green	63	0.21	21	21.4
Red	38	0.2533333	25.3333	Red	25	0.1666667	16.6667	Red	63	0.21	21	21.4
Yellow	15	0.1	10	Yellow	18	0.12	12	Yellow	33	0.11	11	14.3
Total	150	1	100	Total	150	1	100	Total	300	1	100	100

Blue: 47% (Sample) versus 42.9% (Model)

Green: 21% (Sample) versus 21.4% (Model)

Red: 21% (Sample) versus 21.4% (Model)

Yellow: 11% (Sample) versus 14.3% (Model)

8:00 Samples

Color Bowl I - Sampling with Replacement
#1				#2				Pooled 12
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	15	0.3	30	Blue	7	0.14	14	Blue	22	0.22	22	25
Green	15	0.3	30	Green	23	0.46	46	Green	38	0.38	38	35.714286
Red	12	0.24	24	Red	12	0.24	24	Red	24	0.24	24	25
Yellow	8	0.16	16	Yellow	8	0.16	16	Yellow	16	0.16	16	14.285714
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#3				#4				Pooled 34
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	16	0.32	32	Blue	9	0.18	18	Blue	25	0.25	25	25
Green	16	0.32	32	Green	20	0.4	40	Green	36	0.36	36	35.714286
Red	10	0.2	20	Red	11	0.22	22	Red	21	0.21	21	25
Yellow	8	0.16	16	Yellow	10	0.2	20	Yellow	18	0.18	18	14.285714
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
#5				#6				Pooled 56
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	14	0.28	28	Blue	12	0.24	24	Blue	26	0.26	26	25
Green	14	0.28	28	Green	20	0.4	40	Green	34	0.34	34	35.714286
Red	13	0.26	26	Red	11	0.22	22	Red	24	0.24	24	25
Yellow	9	0.18	18	Yellow	7	0.14	14	Yellow	16	0.16	16	14.285714
Total	50	1	100	Total	50	1	100	Total	100	1	100	100
Pooled 135				Pooled 246				Pooled All
Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Color	Count	Proportion	Percent	Truth
Blue	45	0.3	30	Blue	28	0.1866667	18.667	Blue	73	0.243333	24.333	25
Green	45	0.3	30	Green	63	0.42	42	Green	108	0.36	36	35.714286
Red	35	0.2333333	23.333	Red	34	0.2266667	22.667	Red	69	0.23	23	25
Yellow	25	0.1666667	16.667	Yellow	25	0.1666667	16.667	Yellow	50	0.166667	16.667	14.285714
Total	150	1	100	Total	150	1	100	Total	300	1	100	100

Blue: 24.3% (Sample) versus 25% (Model)

Green: 36% (Sample) versus 35.7% (Model)

Red: 23% (Sample) versus 25% (Model)

Yellow: 16.7% (Sample) versus 14.3% (Model)

Some Formulas – Proportions, Percentages, Counts

The class represents some property or attribute, for example, blue, or red. Each member, or unit, of a sample can be classified – the result of the classification of the unit is the unit’s class.

Sample Proportion (p)

n_class ~ number of units of sample in class

n_total ~ total number of units in sample

p_class = n_class / n_total

p_class ~ proportion of sample in class

Sample Percent (pct)

n_class ~ number of units of sample in class

n_total ~ total number of units in sample

p_class = n_class / n_total

pct_class = 100*(n_class / n_total)

pct_class = 100* p_class

pct_class ~ percent of sample in class

Population Proportion (P)

N_class ~ number of units of population in class

N_total ~ total number of units in population

P_class = N_class / N_total

P_class ~ proportion of population in class

Population Percent (PCT)

N_class ~ number of units of population in class

N_total ~ total number of units in population

P_class = N_class / N_total

PCT_class = 100*(N_class / N_total)

PCT_class = 100* P_class

PCT_class ~ percent of population in class

In this setting,

nblue ~ number of blue draws in sample ntotal ~ total number of draws per sample pblue = nblue / ntotal pblue ~ proportion of sample draws showing blue pctblue = 100*pblue pctblue ~ percent of sample draws showing blue Nblue ~ number of blue marbles in bowl Ntotal ~ total number of marbles in bowl Pblue = Nblue / Nblue Pblue ~ proportion of marbles in bowl that are blue

ngreen ~ number of green draws in sample ntotal ~ total number of draws per sample pgreen = ngreen / ngreen pgreen ~ proportion of sample draws showing green pctgreen = 100*pgreen pctgreen ~ percent of sample draws showing green Ngreen ~ number of green marbles in bowl Ntotal ~ total number of marbles in bowl Pgreen = Ngreen / Ngreen Pgreen ~ proportion of marbles in bowl that are green

nred ~ number of red draws in sample ntotal ~ total number of draws per sample pred = nred / nred pred ~ proportion of sample draws showing red pctred = 100*pred pctred ~ percent of sample draws showing red Nred ~ number of red marbles in bowl Ntotal ~ total number of marbles in bowl Pred = Nred / Nred Pred ~ proportion of marbles in bowl that are red

nyellow ~ number of yellow draws in sample ntotal ~ total number of draws per sample pyellow = nyellow / nyellow pyellow ~ proportion of sample draws showing yellow pctyellow = 100*pyellow pctyellow ~ percent of sample draws showing yellow Nyellow ~ number of yellow marbles in bowl Ntotal ~ total number of marbles in bowl Pyellow = Nyellow / Nyellow Pyellow ~ proportion of marbles in bowl that are yellow

The True State of the Bowl

The 6:30 Bowl

Color	Count	Proportion	Percent
Blue	12	12/28 = 0.4285714	42.8571
Green	6	6/28 = 0.2142857	21.4286
Red	6	6/28 = 0.2142857	21.4286
Yellow	4	4/28 = 0.1428571	14.2857
Total	28	1	100

The true proportions are probabilities:

Pr{Blue Shows} =

In long runs of draws with replacement from the bowl, approximately 42.8 percent of draws with replacement from the bowl show blue.

Pr{Green Shows} =

In long runs of draws with replacement from the bowl, approximately 21.4 percent of draws with replacement from the bowl show green.

Pr{Red Shows} =

In long runs of draws with replacement from the bowl, approximately 21.4 percent of draws with replacement from the bowl show red.

Pr{Yellow Shows} =

In long runs of draws with replacement from the bowl, approximately 14.3 percent of draws with replacement from the bowl show yellow.

The 8:00 Bowl

Color	Count	Proportion	Percent
Blue	7	7/28 = 0.25	25
Green	10	10/28 = 0.3571429	35.714
Red	7	7/28 = 0.25	25
Yellow	4	4/28 = 0.1428571	14.286
Total	28	1	100

The true proportions are probabilities:

Pr{Blue Shows} =

In long runs of draws with replacement from the bowl, approximately 25 percent of draws with replacement from the bowl show blue.

Pr{Green Shows} =

In long runs of draws with replacement from the bowl, approximately 35.7 percent of draws with replacement from the bowl show green.

Pr{Red Shows} =

In long runs of draws with replacement from the bowl, approximately 25 percent of draws with replacement from the bowl show red.

Pr{Yellow Shows} =

In long runs of draws with replacement from the bowl, approximately 14.3 percent of draws with replacement from the bowl show yellow.

We see reasonable, but not exact matches between the sample proportions and the probabilities (P).

6:30

Blue: 47% (Sample) versus 42.9% (Model)

Green: 21% (Sample) versus 21.4% (Model)

Red: 21% (Sample) versus 21.4% (Model)

Yellow: 11% (Sample) versus 14.3% (Model)

8:00

Blue: 24.3% (Sample) versus 25% (Model)

Green: 36% (Sample) versus 35.7% (Model)

Red: 23% (Sample) versus 25% (Model)

Yellow: 16.7% (Sample) versus 14.3% (Model)

We didn’t get to these, so read up on Ellsberg I and Ellsberg II.

Regarding Ellsberg I

The 1^st Game: The first bowl is 50%/50% split between blue and green. The best we can do is break even, regardless of strategy. The simplest strategy involves picking one of the colors and always betting on that color.

The 2^nd Game: The second bowl is an unknown composite of red and yellow. We might be able to win this game if 1) there is a dominant color and 2) we can determine that dominant color. A simple strategy here is to pick one color and ride it for awhile. Then stop betting and check the number of winning bets. If the color being betted is losing on a regular basis, switch colors.

The 3^rd Game: This game only makes sense if the second bowl is dominant in red, bet on red – if red consistently shows, stay on the second bowl. Otherwise, either stop playing, or stick with the first bowl.

Regarding Ellsberg II

The 1^st Game: The first bowl is 20% red / 40% black / 40% white. The simplest strategy involves picking one of the colors and always betting on that color. Regardless of betting choice, there is a 40% chance of losing for the single bet, and 20% for getting kicked off the game.

The 2nd Game: The second bowl is 20% red / 80% black or white. The simplest strategy involves picking one of the colors and always betting on that color. If either white or black is sufficiently dominant, this game might be worth playing. The problem is that regardless of the possible advantage in the white/black part of the bowl, there is still a 20% chance of getting killed (permanently losing). But to detect this advantage, one is forced to pick a betting color (white or black) and spend some money.

The idea underlying the Ellsberg games is to illustrate the concept of making decisions about selected processes or populations by making decisions using random samples.