# Sampling from a Box of Two Types of Colored Balls

Here is a standard sampling-without-replacement problem. Suppose we have $$N$$ balls in a box, $$B$$ that are black and the remaining $$W = N - B$$ balls are white.

You sample $$n$$ balls from the box and $$b$$ turn out to be black. What have you learned about the number $$B$$ that are black in the box?

This is a Bayes’ rule problem. We’ll illustrate it for the case where $$N = 50$$.

1. (Prior) First the number of black balls in the box $$B$$ could be any value from 0 to 50. We place a uniform prior on the values 0, 1, 2, …, 50.
library(TeachBayes)
bayes_df <- data.frame(B=0:50, Prior=rep(1/51, 51))
1. (Data) We take a sample of $$n = 10$$ balls without replacement and observe that the number of black balls is $$b = 3$$. The likelihood is the probability of this outcome, expressed as a function of $$B$$. I use the special function dsampling().
sample_b <- 3
pop_N <- 50
sample_n <- 10
bayes_df$Likelihood <- dsampling(sample_b, pop_N, bayes_df$B, sample_n)
1. Last we turn the Bayesian crank using the bayesian_crank() function and obtain the posterior probabilities for $$B$$.
bayes_df <- bayesian_crank(bayes_df)

I compare the prior and posterior probabilities for $$B$$ graphically.

prior_post_plot(bayes_df)

Here is a 90 percent probability interval for $$B$$:

library(dplyr)
discint(select(bayes_df, B, Posterior), 0.90)
## $prob ## [1] 0.9125933 ## ##$set
##  [1]  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26