Probability - Binomial Theorem

Introduction

• The Binomial Theorem can be used to find probability.
• It is commonly used when the number of experiments is large.
• In this method, you don't have to list the sample space of the experiment.
• This method helps you calculate the probability that an event 'a' will occur a specific number of times when the experiment is repeated a specific number of times.
• For example, if a coin is tossed 1000 times, using the Binomial Theorem, you can calculate the probability that P(heads) will occur exactly 450 times.

Formula

`P(x) = ^nC_r(a)^x(1-a)^{n-x}`

In the above formula,

• 'a' is the probability of occurrence of the favorable event (given in the question)
• '1 - a' means the probability of non-occurrence of the favorable event
• 'n' is the number of times the experiment is repeated
• P(x) means the probability of an event occurring 'x' times.

Use

The binomial theorem helps you to calculate probability that an event 'a' will occur a given number of times when an experiment (that can result in 'a') is performed a given number of times. For example, if you toss a coin a thousand times, the binomial theorem can help you get the probability of getting exactly 470 heads out of 1000 tosses.