Pólya's Experiment

Ball 1 Ball 2 Ball 3 Ball 4 Ball 5 Ball 6 Ball 7 Ball 8 Ball 9 Ball 10 Ball 11 Ball 12 Ball 13 Ball 14 Ball 15 Ball 16 Ball 17 Ball 18 Ball 19 Ball 20 Ball 21 Ball 22 Ball 23 Ball 24 Ball 25 Ball 26 Ball 27 Ball 28 Ball 29 Ball 30 Ball 31 Ball 32 Ball 33 Ball 34 Ball 35 Ball 36 Ball 37 Ball 38 Ball 39 Ball 40 Ball 41 Ball 42 Ball 43 Ball 44 Ball 45 Ball 46 Ball 47 Ball 48 Ball 49 Ball 50
Distribution graph

Description

This app models Pólya's urn experiment. An urn initially contains \(a\) red and \(b\) green balls. On each trial, a ball is selected at random from the urn, and then replaced along with \(c\) new balls of the same color. In this simulation, the red balls are numbered consecutively \((1, 2, \ldots)\) and the green balls are numbered consecutively in the same manner. The sample of balls in \(n\) trials is shown in the left graph box. Random variable \(Y\) gives the number of red balls selected after \(n\) trials, and \(M\) gives the proportion of red balls selected in the \(n\) trials. These are recorded on each update in the data table.

The probability density function and moments of \(Y\) are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density function and moments of \(Y\) are shown in red in the distribution graph and are recorded in the distribution table. The parameters \(a\), \(b\), \(c\), and \(n\) can be varied with the input controls.