Here, you will learn formula for binomial probability distribution in probability with example.
Let’s begin –
Suppose that we have an experiment such as tossing a coin or die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a red marble. In some cases this probability will not change from one trial to the next(as in tossing a coin or die). Such trials are then said to be independent and are often called Bernoulli trials after James Bernoulli who investigated them at the end of the seventeenth century.
Formula for Binomial Probability Distribution
Let p be the probability that an event will happen in any single Bernoulli trial(called the probability of success).Then q = 1 – p is the probability that the event will fail to happen in any single trial (called the probability of Failure). The probability that the event will happen exactly x times in n trials (i.e., x success and n – x failures will occur) is given by the probability function.
f(x) = P(X = x) = \(\binom{n}{x} p^x q^{n-x}\) = \(n!\over {x!(n – x)!}\) \(p^xq^{n-x}\)
where the random variable X denotes the number of success in n trials and x = 0, 1,…….,n.
Example : What is the probability of getting exactly 2 heads in 6 tosses of a fair coin?
Solution : The probability of getting exactly 2 heads in 6 tosses of a fair coin is
P(X = 2) = \(\binom{6}{2} ({1\over 2})^2 ({1\over 2})^{6-2}\)
= \(6!\over {2!4!}\) \(({1\over 2})^2 ({1\over 2})^{6-2}\)
= \({15}\over{64}\)
The discrete probability function is often called the binomial distribution since for x = 0, 1, 2,……,n, it corresponds to successive terms in the binomial expansion.
\((q + p)^n\) = \(q^n\) + \(\binom{n}{1} p q^{n-1}\) + \(\binom{n}{2} p^2 q^{n-2}\) + ……….+ \(p^n\) = \({\sum_{n=1}^{\infty}}\)\(\binom{n}{x} p^x q^{n-x}\)
