Building a single likelihood for the sum of two events Consider an unknown number of adult deer, $N$, which can be split into two groups - males, $M$, and females, $F$, where $N = F + M$. These adults were hunted during the year (a lethal form of sampling without replacement), with each group having its own individual-level probability of being killed by a hunter, $p_f$ and $p_m$, respectively. For example, male deer in Michigan have a 40% of being killed by a hunter during the hunting season while females only have a 16% chance of being killed. The number of adults that were killed can be denoted as $m$ and $f$, respectively. 
I want to build a likelihood for the total number of adults, $N$, given that I know $p_m$, $p_f$, $m$, and $f$, to be used in a larger joint likelihood. If I were interested in building a likelihood for the two different groups of adults, I could write:
$$L(M \mid p_m,m) = \binom{M}{m} \cdot (p_m)^{m} \cdot (1-p_m)^{M-m}$$
$$L(F \mid p_f,f) = \binom{F}{f} \cdot (p_f)^{f} \cdot (1-p_f)^{F-f}$$
What I want, however, is
$$L(N \mid p_m,p_f,m,f) = \text{?}$$
The full likelihood already has a large number of parameters to estimate, so I'd rather just have a single parameter for the total number of adults, as opposed to having a separate parameters to estimate for males and females. Basically, the likelihood equation cannot contain $M$ or $F$.
Any help would be greatly appreciated!
 A: After discussing this with some colleagues, the (sort-of) solution turns out to be to use a multinomial:
$$L(N \mid p_m,p_f,m,f) = \binom{N}{m,f} \cdot (p_m)^{m} \cdot (p_f)^{f} \cdot (p_{bar})^{N-m-f},$$
where $p_{bar}$ is the probability of not being killed by a hunter. This probability is unknown, but can be calculated as the probability of not being hunted given that you are a male times the probability that you are a male plus the probability of not being hunted given that you are a female times the probability that you are a female.
$$p_{bar} = P(not.hunted|male) \cdot P(male) + P(not.hunted|female) \cdot P(female)$$
$$p_{bar} = (1-p_m) \cdot P(male)+(1-p_f) \cdot P(female)$$
Unfortunately, we do not know the probability of being a male or female, so we are left with an unknown quantity - the ratio of males in the population, $R$:
$$p_{bar} = (1-p_m) \cdot R + (1-p_f) \cdot (1-R)$$
And the only way to calculate $R$ is to know either $M$ or $F$. So the short of it is, as far as I can tell, there is no way how to eliminate both $M$ and $F$ from the likelihood.
