In this post, we’ll discuss Binomial Random Variables.
Prerequisite : Random Variables
A specific type of discrete random variable that counts how often a particular event occurs in a fixed number of tries or trials.
For a variable to be a binomial random variable, ALL of the following conditions must be met:
- There are a fixed number of trials (a fixed sample size).
- On each trial, the event of interest either occurs or does not.
- The probability of occurrence (or not) is the same on each trial.
- Trials are independent of one another.
n = number of trials p = probability of success in each trial k = number of success in n trials
Now we try to find out the probability of k success in n trials.
Here the probability of success in each trial is p independent of other trials.
So we first choose k trials in which there will be success and in rest n-k trials there will be failure. Number of ways to do so is
Since all n events are independent, hence the probability of k success in n trials is equivalent to multiplication of probability for each trial.
Here its k success and n-k failures, So probability for each way to achieve k success and n-k failure is
Hence final probability is
(number of ways to achieve k success and n-k failures) * (probability for each way to achieve k success and n-k failure)
Then Binomial Random Variable Probability is given by:
Let X be a binomial random variable with the number of trials n and probability of success in each trial be p.
Expected number of success is given by
E[X] = np
Variance of number of success is given by
Var[X] = np(1-p)
Example 1 : Consider a random experiment in which a biased coin (probability of head = 1/3) is thrown for 10 times. Find the probability that the number of heads appearing will be 5.
Let X be binomial random variable with n = 10 and p = 1/3 P(X=5) = ?
Here is the implementation for the same
Probability of 5 heads when a coin is tossed 10 times where probability of each head is 0.333333 is = 0.136565
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