Calculating Poisson process problem

Question

A shop sells both hot and cold drinks. Hot drink sales occur at the instants of a Poisson process with expectation 30 drinks per hour.Cold drink sales occur at the instants of a Poisson process with expectation 20 drinks per hour. 60% of customers purchasing a drink are female, 40% of customers purchasing a drink are male.

The Question is: exactly four cold drinks are sold to men during the third hour of the day The standard answer is standard answer

I understand this answer but I just tried a different approach to this question and got a different answer which is: let x=the number of male customers who bought cold drink and m=total cold drink sales. my answerwhich is a different answer compared to the standard answer, where did I go wrong?

Answer 1

I think the original problem should be

What is the probability that exactly four cold drinks are sold to men during the third hour of the day?

Well, cold drinks sales follow a Poisson process with rate 20. It is given that 40% of customers are male. Then by Poisson thinning, cold drinks sales to men follows a Poisson process with rate $.4\cdot 20=8$. This means that the number of cold drinks sold to men in $t$ hours $N$ follows $\text{Pois}(8t)$. Thus, the third hour is just $t = 1$, and $$P(N = 4) = \frac{e^{-8}(8)^4}{8!}.$$ You don't know how many drinks were sold so you cannot say $m = 4$. You also cannot say $x = 4$ because you don't know how many drinks were sold to men in any case. You are trying to calculate the chance that the number of cold drinks sold to men is 4. It is not given.