Can someone explain to me why combination is used here? I don’t quite understand how this work.
In: Mathematics
Let r = 2 and n = 4.
There are six ways in which two out of the four trials can be successful:
First and second
First and third
First and fourth
Second and third
Second and fourth
Third and fourth
Hence, 4!/(2!2!) = 6.
Make sense?
The binomial distribution tells you, if you have a (possibly biased) coin with probability p of showing head, then throw it n times, what is the probability for each possible number of head showing up.
There are 2^n outcomes. You can list out all 2^n of them by writing down a sequence of n letters, each being either H or T.
Among those, nCr outcomes give you the same number r of head showing up. Each outcome can be uniquely identified by choosing r positions in the sequence of n, fill those with H, then make everything else T. Since there are nCr ways of picking r out of n, there are nCr such outcomes.
The probability of each outcome is exactly p^r q^(n-r) , because for that possibility to show up when you throw your coin, the head need to show up whenever the sequence said H, the tail need to show up whenever the sequence said T, and there cannot be any mistake. So if you’re at the point when you need head, the probability that head show up is only p, and when you’re at the point when you need tail, the probability that tail show up is only q=1-p. Since head need to happen r times, tail need to happen n-r times, multiplying up gives you the probability of this outcome, which is p^r q^(n-r).
Adding up the probability of all outcome with the same number of head, you get nCr p^r q^(n-r) which is the binomial distribution.
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