09 Urn Problem

**An urn contains black and white balls, in unknown proportion. You pull $n$ balls with replacement. Observed sequence : $x=BBWBWWBWWWBW$

Question

What is the true proportion of black balls in the urn that best explains this sequence?

• Let the probability of drawing a black ball : $\theta$
• Since there are 2 types of balls, the probability follows a Bernoulli distribution with $P(B)=\theta$.
• Hence, the probability of drawing a white ball is $P(W)=1-\theta$.
• The urn’s content doesn’t change and we draw with replacement, so the draws are i.i.d. (independent and identically distributed).
• So, given $\theta$, the probability of the sequence is

$$P(x;\theta )=\prod _{i=1}^{n}P(x_i;\theta )$$

• The likelihood function is $P(x;\theta )$ viewed as the function of $\theta$:

$$L(\theta \mid \mathbf{x})=\prod _{i=1}^n\{\begin{array}{ll}\theta ,& \text{if }{x}_i=B\\ 1-\theta ,& \text{if }{x}_i=W\end{array}$$

• Likelihood measures how well $\theta$ fits the data in the Bernoulli model.