Rolled a 3 from a randomly choosing betwen 6 and 4 sided dice. What is the probability the dice was 4 sided

Question

I can't determine if this is simply a conditional probability with the probability of rolling a 3 from a 4 sided dice being 1/2*1/4, or if this may actually require a Bayesian approach.

Answer 1

You correctly realized that you are asked the conditional probability of having picked the 4 sided die given that you rolled a $3$, but the $\frac{1}{2}\cdot \frac{1}{4}$ is not a conditional probability. Rather, that's the probability of rolling a 3 and picking the 4 sided dice.

That is, you computed $P(D4 \cap R3)$, but you want to compute $P(D4 | R3)$ where

$D4$ : event of picking $4$ sided die

$R3$ : event of rolling a $3$

And for that, indeed you should use the Bayesian rule.

I also would like to point out that your answer of $\frac{1}{8}$ should have pretty obviously been incorrect, for that would mean that there was a $\frac{7}{8}$ chance that you had picked the 6 sided die given a roll of 3, i.e. that the chance of having picked the latter die was seven times the chance of having picked the former?! Given that the two dice have a similar number of sides, the chance should be much closer to $\frac{1}{2}$ for wither one. So, getting $\frac{1}{8}$ should have set off the alarm bells right away.

In fact, given that the 4 sided die has fewer sides, the chance of having picked that one given a roll of 3 should be a little over $\frac{1}{2}$: think of the extreme situation where you have a 1 sided die whose only side is a 3, and a 1000 sided die only one of which is a 3. Now, if yu roll a 3, which die is more likely to have been used? The 1 sided die of course, since it is more likely to come up with a 3 than the 1000 sidd die. Well, likewise, the 4 sides die is more likely to get a 3 than the 6 sided die, and therefore, getting a 3 means that you more likely used the 4 sided die than the 6 sided die. But given that the 4 sided die is not all that more likely to get a 3 than the 6 sided die, it follows that having rolled a 3, you are only slightly more likely to have used the 4 sided die than the 6 sided die, meaning that the answer should be a little over $\frac{1}{2}$

The moral is: always consider your answer and see if it makes intuitive sense!