Can we find the probability by counting the outcomes? A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?
We can solve this problem by applying the law of total probability and the answer is 19/42.
My question is: can we solve this problem by counting the total number and favourable number of outcomes? If yes how and if no why?
 A: The main problem is that to get the correct probabilities with counting favourable/all outcomes, your outcomes need to all have the same probabilities. This isn't true in this problem, so the counting technuique will not work in this example.
The "not-mentioned information" in the problem description that has to be inferred from the text is that you do two random choices:

*

*You choose the purse at random first (each with probability $0.5$).

*You then choose a coin at random (uniform distribution) from all the coins in the selected purse.

Because of that, the probability that any specific coin from purse 1 is selected is $\frac12\times\frac16=\frac1{12}$, because you need to select purse 1 in step 1 above (probability is $\frac12$) and then choose that specific coin from the 6 coins in that purse in step 2.
Similiarly, the probability to select any specific coin from purse 2 is $\frac12 \times \frac17 = \frac1{14}$.
Notably, the probabilities are different. This means that the counting technique of

*

*13 coins in total that can be selected and

*6 of those are silver coins

leads to the incorrect result that the asked for probability is $\frac6{13}$.
If we distribute the coins more extremely, the differences between the approaches becomes more obvious.
Say purse 1 comes from a medivial beggar and contains just 1 copper coin, while purse 2 comes from a nobleman and contains 100 silver coins.
If we ask the question again, what is the probability to select a silver coin from a randomly selected purse, it becomes clear that step 1, (choose a purse at random) is the all important one, because step 2 (choose a coin from the purse at random) is not important at all.
So the answer in this case is simply $p=\frac12$. This is in stark contrast to the counting method, which says we have 101 coins, 100 of which are silver. The reason is that choosing a specific silver coin is very unlikely ($\frac1{200}$), while slecting the one copper coin has probability $\frac12$.
So if you want to apply the counting method and use the formula that the probability is favourable outcomes divided by all outcomes, you need to make absolutely sure that all the outcomes you are counting have the same probability.
