My friends asked me two Bayes questions a while ago. Due to the rapid spread of COVID-19 in US, I’ve been at home for five days. So I decide to put my solutions here for references. The first question is as follows.

There are five balls in a bag. Each ball is colored either black or white with equal chance. Now suppose that someone have already drawn the ball 4 times with replacements, among which he/she got 1 white ball and 3 black balls as a result. Find the probability distribution of the number of white balls in that bag.

This problem is a straightforward application of Bayes’ Rule. The only tricky part is to formulate the problem statement with math language.

Let $X$ be the number of white balls and let event $A = [\text{1 white ball and 3 black balls among 4 trials}]$. The prior distribution of $X$ is simply the binomial distribution $B(5, \frac{1}{2})$. Actually, we are interested in finding the posterior probability $P(X=x\vert A)$. Note that

\[P(A\vert X=x) = \begin{cases} b(1; 5, \frac{x}{5}),~~~& x= 1, 2,3 \text{ or }4\\ 0,~~~& x=0 \text{ or } 5 \end{cases},\]where $b(x;n,p)$ is the pmf at $x$ for a $B(n, p)$ r.v.

We use Bayes’s Rule as follows:

\[\begin{aligned} P(X=x\vert A) &= \frac{P(X=x, A)}{P(A)} = \frac{P(A\vert X=x)P(X=x)}{\sum_i P(A|X=i)P(X=i)} \\ & = \frac{ {4 \choose 3} \cdot \left(\frac{x}{5}\right)^1\cdot \left(\frac{5-x}{5} \right)^3 \cdot {5 \choose x}\cdot\left(\frac{1}{2}\right)^5 } {\sum\limits_{i=1}^4 {4 \choose 3} \cdot \left(\frac{i}{5}\right)^1\cdot \left(\frac{5-i}{5} \right)^3 \cdot {5 \choose i}\cdot\left(\frac{1}{2}\right)^5},~~~ x=1,...,4. \end{aligned}\]I used R to compute the above quantity:

```
> x <- rep(0,4)
> for (i in 1:4){
+ x[i] <- 4 * (i/5) * ((5-i)/5)^3 * choose(5, i) / 2^5
+ }
> y <- x/sum(x)
> y
[1] 0.28571429 0.48214286 0.21428571 0.01785714
```

Thus, we have

\[P(X=x\vert A) = \begin{cases} 0.28571429,~~~& x= 1\\ 0.48214286,~~~& x= 2\\ 0.21428571,~~~& x= 3\\ 0.01785714,~~~& x= 4\\ 0,~~~& x=0 \text{ or } 5 \end{cases}.\]When we are asked to give a point estimate of the number of white balls, we typically use the posterior mode, which is also known as the MAP (maximum a posteriori) estimation. In this case, it is $2$.