Estimating probabilities using Bayes rule? I am working on a past exam paper. I am given a data set as follows:


*

*Hair: {brown, red} = $\{B,R\}$

*Height: {tall, short} = $\{T,S\}$

*Country: {UK, Italy} = $\{U,I\}$


Our sample is:
$$(B,T,U)\quad (B,T,U)\quad (B,T,I)\quad (R,T,U)$$
$$(R,T,U)\quad (B,T,I)\quad (R,T,U)\quad (R,T,U)$$
$$(B,T,I)\quad (R,S,U)\quad (R,S,U)\quad (R,S,I)$$
Question. Estimate the probabilities $P(B,T\mid U)$, $P(B\mid U)$, $P(T\mid U)$, $P(U)$ and $P(I)$.
As the question states estimate, I am guessing that I don't need to calculate any values. Is it just a case of adding up how many times $P(B,T\mid U)$ occurs over the whole data set e.g. $(2/12) = 16\%$.
Then would the probability of $P(U)$ be $0$?
 A: If I've read the question correctly, then I think your reasoning is sound, but your numbers are a little off.
If we assume that the probabilities associated with hair colour, height and country are all independent, then I think your approach of just counting works reasonably well. The numbers just need adjusting:
$$P(B,T \mid U) = \frac{\text{number of events of the form $(B,T,U)$}}{\text{number of events with country } U} = \frac{2}{8} = \frac{1}{4}.$$
Notice that the denominator is $8$, not $12$ as you stated. We're only looking at events where it's given that the country is $U$, so we don't count events where the country is $(\text{not } U)$.
Repeat similarly as appropriate for the other events.

Collected other thoughts:
We can easily see that $P(U)$ is probably non-zero, since we have people whose country is $U$. If $P(U)=0$, then this would clearly be ridiculous.
I'm not sure how much use Bayes' rule is here. I can see why you might want to use it (and maybe there's a way to apply it that I've missed), but you'll still need to calculate conditional probabilities from the sample somewhere, so you might as well go straight to it.
Finally, as Henning noted in a comment, the word "estimate" has a different meaning in statistics. Normally it's just a fancy word for a guess, but in statistics it really means an educated guess of some quantity (such as the probability distribution, which we're calculated here) based on some data we've collected.
For example, if I count the number of red cars and blue cars on a road (say, 10 and 20, respectively), then I can estimate the probability distribution as
$$P(\text{colour} = \text{red}) = \frac{10}{10+20} = \frac{1}{3}, \qquad P(\text{colour} = \text{red}) = \frac{20}{10+20} = \frac{2}{3}.$$
I've still done numerical calculations, but this is still called an estimate in statistics.
