# Non-technical question about maximum likelihood estimation / intuition

Just a quick question on MLE, sorry if its too basic but I would love help on this issue!

If we want to calculate the probability $P$ of the number of males in the United States and we knew the distribution $pmf$/$pdf$, why can't we just take the arithmetic mean and use $\bar{y}$. Why go through the trouble of using the MLE?

For example, if we suppose that we are looking at a Bernoulli R.V. where $X_1=1$ is male and $X_2=0$ is a female and we want to estimate the parameter, which is $p$ in the case of Bernoulli, why would we go to great lengths to find the MLE when we could just see the $mean$ from our sample?

I'm very confused on why MLE is useful in that way especially when we have the data. Thanks so much!

Say you have a RV that is uniform in $[0,A]$ and you want to estimate $A$. Suppose that we take four sample values and they are $1$, $2$, $3$ and $34$. The mean of your sample is $10$. We also know that uniform RV in $[0,A]$ has a mean of $A/2$. So blindly trusting the mean will lead you to believe that $A=20$ which is absurd because we have the sample $34$. So mean is not a good indication at all.
It turns out that $MLE$ is not a good estimate also! If $A<34$, then the probability that we will see $34$ is zero. If $A>34$ then the probability that all the samples are less than $34$ decreases as $A$ increases. So the most probable value (MLE) is $A=34$. This turns out to underestimate $A$!
The above problem has a very interesting history. During world war II, the British knew that Germans would assign serial numbers to the tanks. So the serial numbers come from a uniform distribution between $1$ and $A$ where $A$ is the number produced. So based on the tanks that were captured/destroyed, the British had random samples from the uniform distribution. They used the unbiased estimate of $A$ to get very accurate estimate of the number of tanks in the German army. Look at http://en.wikipedia.org/wiki/German_tank_problem