I am reviewing the basic concepts of Calculus, Linear Algebra & Probability and Statistics for Graduate Courses. I learned many of these courses back in high school quite a while back and they are a bit rusty. While revising Probability part I came across one particular question that is confusing to me and I couldn't find relevant information.
So the question is that a coin x is tossed 5 times and following sample is observed: {1, 1, 0, 1, 0} where 1 = head ; 0 = tail
The question related to this data are firstly find the probability of observing this data. This part is simple since all possible spaces can be 2^5 = 32 and this particular one has a probability of 1/32 right? (assuming that the question is asking for this particular setting and not that there should be three heads and 2 tails). Any comments?
The second part is what is confusing it says that since the probability distribution is equal we got above probability, but if the probability of getting heads was not 0.5 the probability observed in previous question would change and I am supposed to find a distribution that maximizes the probability of the sample. I am confused? Does probability distribution effect the observed sample's probability? If so then how can this be solved? Can anyone point me to some material that I can review to refresh my knowledge?
This can be solved by setting variable $p$ as probability on e.g. a head.
Then the probability of observing this data is $p^3(1-p)^2$ and you are asked to find the maximal value of this polynomial in $p$ under the extra condition that $p\in[0,1]$.
So at this point we somehow leave "probability theory" and enter "calculus".
Can you do that yourself?