Binomial Random Variable – and Probability of SINGLE event happening more than once (multiple heads in a coin-toss etc.)
Whenever you have a variable that can take on two possible values (flip of a coin for example), you can calculate the probability for a ‘succession’ of events using the individual event probabilities. for e.g. if ‘p’ is the probability for a ‘heads’ – then the probability of getting n heads in a total of ‘m’ flips is
p raised to n TIMES (1 – p) raised to m-n
However – the number of ways you can arrive at ‘n’ heads is more than one – (e.g. first n tosses all are heads, every alternate toss is a head etc.). To account for this, we multiply by the number of ways you can get ‘n’ heads in ‘m’ tosses – which is m C n (m Combinatorics n)
The total probability of getting ‘n’ heads in a total of ‘m’ tosses then is:
(mCn) * p^n * (1-p) ^ (m-n)
This is nothing but the Mth ‘binomial coefficient’ in the expansion of (p + 1-p) raised to n. To verify this, simply take the sum of all co-efficients
SUM (p + 1-p) ^ n = 1 ^ n = 1
The net probability remains 1.
It is amazing how the binomial theorem ties into ‘two-valued’ event probabilities.