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 ?, 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:
Xi = {1 if the ith ball is blue, 0 if the ith ball is red
Note that the Xi's are i.i.d and Xi ~ Bernoulli(?/3).
After doing my experiment, I observe the
following values for Xi's.
x1 = 1, x2 = 0, x3 = 1 and x4 = 1
(i) For each possible value of ?, find the
probability of the observed sample.
(ii) For which value of ? is the probability of the
observed sample the largest?