Using a "population" consisting of probabilities to predict accuracy of sample. Since I'm not sure if the title explains my question well enough I've come up with an example myself: Let's say I live in a country where every citizen goes to work everyday and every citizen has the choice to go by bus or by train (every citizen makes this choice everyday again - there are almost no citizens who always go by train and never by bus, and vice-versa).

Using a "population" consisting of probabilities to predict accuracy of sample
Since I'm not sure if the title explains my question well enough I've come up with an example myself:
Let's say I live in a country where every citizen goes to work everyday and every citizen has the choice to go by bus or by train (every citizen makes this choice everyday again - there are almost no citizens who always go by train and never by bus, and vice-versa).
I've done a lot of sampling and I have data on one million citizens about their behaviour in the past 1000 days. So, I calculate the "probability" per citizen of going by train on a single day. I can also calculate the average of those calculated probabilities of all citizens, let's say the average probability of a citizen going by train is 0.27. I figured that most citizens have tendencies around this number (most citizens have an individual probability between 0.22 and 0.32 of going by train for example).
Now, I started sampling an unknown person (but known to be living in the same country) and after asking him 10 consecutive days whether he went by train or by bus, I know that this person went to his work by train 4 times, and by bus 6 times.
My final question: how can I use my (accurate) data on one million citizens to approximate this person's probability of going by train?
I know that if I do the calculation the other way around, so, calculate the probability of this event occurring given the fact that I know this person's REAL probability is 0.4 this results in: ${0.4}^4\cdot <!--\; \cdot \; -->{0.6}^6\cdot <!--\; \cdot \; -->10C4=\sim <!--\; \sim \; -->25\mathrm{\%<!--\; \%\; -->}$. I could calculate this probability for all possible probabilities between 0.00 and 1.00 (so, $0\mathrm{\%<!--\; \%\; -->}-<!--\; -\; -->100\mathrm{\%<!--\; \%\; -->}$ without any numbers in between) and sum them all, which sums to about 910%. I could set this to 100% (dividing by 9.1) and set all other percentages accordingly (dividing everything by 9.1 - so, our 25% becomes ~2.75%) and come up with a weighted sum: $2.75\mathrm{\%<!--\; \%\; -->}\cdot <!--\; \cdot \; -->0.4+X\mathrm{\%<!--\; \%\; -->}\cdot <!--\; \cdot \; -->0.41$ etc., but this must be wrong since I'm not taking my accurate samples of the population into account.