Prediction and Perfect Samples

Probabilities and Expected Values Consider the color bowl in blue, green, red and yellow yellow. The proportion of each color in the bowl is a probability, and drives the frequency of that color in random samples from the bowl.

Blue Pr{Blue Shows} = N Blue /N Total E Blue = n* Pr{Blue Shows} = n*(N Blue /N Total ) In random samples of size n, we expect E Blue of n draws with replacement to show as blue.

Blue P Blue =Pr{Blue Shows} = N Blue /N Total In long runs of draws with replacement from the bowl, we expect approximately 100*P Blue % of draws to show as blue.

Green Pr{Green Shows} = N Green /N Total E Green = n* Pr{Green Shows} = n*(N Green /N Total ) In random samples of size n, we expect E Green of n draws with replacement to show as green.

Green P Green = Pr{Green Shows} = N Green /N Total In a long runs of draws with replacement from the bowl, we expect approximately 100*P Green % of draws to show as green.

Red P Red = Pr{Red Shows} = N Red /N Total E Red = n*P Red In a random samples of size n, we expect E Red of n draws with replacement to show as red.

Red P Red = Pr{Red Shows} = N Red /N Total In long runs of draws with replacement from the bowl, we expect approximately 100*P Red % of draws to show as red.

Yellow Yellow Yellow Yellow P Yellow = Pr{Yellow Shows} = N Yellow /N Total YellowYellow E Yellow = n* P Yellow Yellow In random samples of size n, we expect E Yellow of n yellow draws with replacement to show as yellow.

Yellow Yellow Yellow Pr{Yellow Shows} = N Yellow /N Total In long runs of draws with replacement from the Yellow bowl, we expect approximately 100*P Yellow % of yellow draws to show as yellow.

Perfect Samples and Prediction If we know the population frequencies for the bowl, we can compute the model and predict the behavior of random samples from the bowl. In n draws with replacement from our bowl, we expect approximately E Blue draws, approximately E Green draws, approximately E Red draws and Yellow approximately E Yellow draws.

The Model and Prediction If we know the population frequencies for the bowl, we can compute the model and predict the behavior of random samples from the bowl. In long runs of draws with replacement from our bowl, we expect approximately 100*P Blue % of draws to show as blue, approximately 100*P Green % of draws to show as green, approximately 100*P Red % of draws to show as red and Yellow yellow approximately 100*P Yellow % of draws to show as yellow.

The Color Bowl - Sampling

First, mix the bowl – randomize the positions of the marbles in the bowl. The idea is to give, on each draw, each marble in the bowl the same chance of being drawn. After mixing the bowl, draw a single marble from the bowl, note the color of that marble, and then return that marble to the bowl. The group will repeat this process 50 times, recording the frequencies of both observed colors.

Division of Labor Mix: Prior to each draw, thoroughly mix the bowl. Draw: Draw a single marble from the bowl, note the color, then return the marble to the bowl. Track: Track the number of draws made and record the frequency of each color.

The Color Bowl Model The frequencies of the colors in the bowl form the model, and drive the occurrence of that color in samples drawn from the bowl. The observed sample frequencies for the colors suggest the possible model frequencies for the bowl. That is, the sample (observed) color frequencies suggest the model or population (expected) color frequencies.