Binomial Distribution


Chance of surviving a trip to Tortula!

Why should you know this distribution?

Back of napkin probability calculations like Betting on flipping a coin, lottery, die throw, etc. where your decisions would need you to have answers to questions like:

  • “Given last 3 flips of this coin were heads what is the probability of next flip getting a tail?”

  • “I have a king and a Queen what is the probability of opponent having 2 aces”

  • “How likely am I, to land 3 six in a row when I roll the die”

Or when you run a production line or a hospital -

  • “My machine produces good products 95% of times, what is the probability of 10 defectives in 1000 samples picked by the inspector”

  • “Sepsis patients have about a 67% chance of going into a shock, I have 56 such patients. I need drug Epinephrine should a patient go into shock but it is a rare medicine, what is the chance of more than 12 patients going into a shock?”

or for more important things in life (at least for me ) - like the choice of weapon in Doom Eternal Boss fight.

  • “Probability of a kill using rocket launcher is 56%, and I have only 3 rockets left. What is the chance of a kill in the first 2 launches.”

Okay, I’m interested — tell me about how does this work?

Let’s start with the slightly technical definition —An X that counts the number of successes in many independent, identical Bernoulli trials is called a binomial random variable. X follows a binomial distribution.

Jargon explained :

Bernoulli trial: Think of this as an event with only 2 outcomes, success or failure, heads or tails, win or no win, a good product or defective, etc.

Independent: One event does not affect the outcome of any other event. ex: the outcome of a previous coin flip in no way affects the outcome of any subsequent flip, Scoring a kill from a bullet does not affect the next bullet scoring a kill or missing the target, etc.

Identical: All the events have the same probability of success ex: every time you flip a coin the prob of landing a tail is 50%, every roll of a die has the same probability of ending in a six.

Now read the definition again — sllloooowly, understanding each word.

Let's take an example: You are about to make a journey & fly through the treacherous nebulae of Tortula to reach your destination. It is famously said that the chance of a blown engine during any ride through this nebulae is 60%.

You have 5 engines your jet and it requires at least 3 engines to be working for the jet to fly. The natural question to ask would be

“What are