I have a bag that contains 3 balls. Each ball is either red or blue, but I have no information in addition to this. Thus, the number of blue balls, call it θ, might be 0, 1, 2, or 3. I am allowed to choose 4 balls at random from the bag with replacement. We define the random variables X₁, X₂, X₃, and X₄ as follows

Note that X_i's are i.i.d. and X_i ∼ Bernoulli(θ/3). After doing my experiment, I observe the following values for X_i's. x₁ = 1,x₂ = 0,x₃ = 1,x₄ = 1. 

Thus, I observe 3 blue balls and 1 red balls.

1. For each possible value of θ, find the probability of the observed sample, (x₁,x₂,x₃,x₄) = (1, 0, 1, 1).

2. For which value of θ is the probability of the observed sample is the largest?

Transcribed Image Text:

X₁ = 1 0 if the ith chosen ball is blue if the ith chosen ball is red