Single Event Probabilities
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.
Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.
Anuj Varma – who has written 1216 posts on Anuj Varma, Hands-On Technology Architect, Clean Air Activist.
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