Calculating Probability of Rolling a Number on a Biased 3-Sided Die

[Kreizhn]

Homework Statement

Suppose we have a biased three sided die. When trying to calculate the probability of rolling a number, we find that half of the time we're accurate, and the other half of the time we observe a random number 1 through 3 (uniformly distributed). I've calculated the probability distribution of observing a given number as $\rho = \begin{pmatrix} p_1 \\ p_2 \\ p_3 \end{pmatrix}$. That is, the probability of rolling "i" is $p_i, i=1,2,3$. Now let's say that in an experiment I throw the three sided die, and a "1" appears. I need to write down the probabilistic state describing my knowledge of how the die lies after the observation.

Homework Equations

Perhaps Baye's law on conditional probability
$$P(a|b) = \frac{P(b|a)P(a)}{P(b)}$$

The Attempt at a Solution

I would imagine this is a one-liner, but I can't quite figure out how to do it.

 

[lippyka]

I feel like the answer would be something like:After observing a "1", the probabilistic state describing my knowledge of how the die lies is \rho = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}.