Binomial Distribution is a Continuous Distribution
Binomial Distribution Formula
The binomial distribution is a commonly used discrete distribution in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment.
Binomial Distribution in Statistics:The binomial distribution forms the base for the famous binomial test of statistical importance. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the boolean-valued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution.
What Is the Binomial Distribution Formula?
The binomial distribution formula is for any random variable X, given by; P(x:n,p) = nC\(_x\) px (1-p)n-x Or P(x:n,p) = nCx px (q)n-x
where,
- n = the number of experiments
- x = 0, 1, 2, 3, 4, …
- p = Probability of success in a single experiment
- q = Probability of failure in a single experiment (= 1 – p)
The binomial distribution formula is also written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x
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Examples on Binomial Distribution Formula
Example 1: If a coin is tossed 5 times, using binomial distribution find the probability of:
(a) Exactly 2 heads
(b) At least 4 heads.
Solution:
(a)The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:
Number of trials: n=5
Probability of head: p= 1/2 and hence the probability of tail, q =1/2
For exactly two heads:
x=2
P(x=2) =5C2 p2 q5-2= 5! / 2! 3! × (½)2× (½)3
P(x=2) = 5/16
(b)For at least four heads,
x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)
Hence,
P(x = 4) =5C4 p4 q5-4 = 5!/4! 1! × (½)4× (½)1 = 5/32
P(x = 5) =5C5 p5 q5-5 = (½)5 = 1/32
Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16
Example 2: For the same question given above, find the probability of getting at most 2 heads.
Solution:
Solution: P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)
P(X = 0) = (½)5 = 1/32
P(X=1) = 5C1 (½)5.= 5/32
Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16
Example 3: 60% of people who purchase sports cars are men. Find the probability that exactly 7 are men if 10 sports car owners are randomly selected.
Solution:
Let's Identify 'n' and 'X' from the problem.
The number of sports car owners are randomly selected is n = 10, and
The number to find the probability is X = 7.
Given: p = 60%, or 0.6.
Therefore, the probability of failure is q = 1 – 0.6 = 0.4
Now, using the binomial distribution formula
\( P( x ) = \frac{{n!}}{{( {n - x} )!x!}}.( p )^x .( q)^{n - x} \\
= \frac{{10!}}{{( {10 - 7} )!7!}}.( {0.6} )^7 .t( {0.4})^{10 - 7} \\
= 120 \times 0.0279936 \times 0.064 \\
= 0.215 \)
Answer: The probability that exactly 7 are men is 0.215 or 21.5%.
FAQs on Binomial Distribution Formula
What Is Binomial Distribution and Binomial Distribution Formula in Statistics?
The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial. The binomial distribution formula is for any random variable X, given by; P(x:n,p) = nC\(_x\) px (1-p)n-x Or P(x:n,p) = nC\(_x\) px (q)n-x, where, n is the number of experiments, p is probability of success in a single experiment, q is probability of failure in a single experiment (= 1 – p) and takes values as 0, 1, 2, 3, 4, …, n.
What Is the Purpose of the Binomial Distribution Formula?
The binomial distribution formula allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.
What Is the Formula for Binomial Distribution?
The formula for binomial distribution is:
P(x: n,p) =nC\(_x\) px (q)n-x
Where p is the probability of success, q is the probability of failure, n = number of trials.
What Is the Binomial Distribution Formula for the Mean and Variance?
The mean and variance of the binomial distribution are:
Mean = np
Variance = npq
where p is the probability of success, q is the probability of failure, n = number of trials.
Source: https://www.cuemath.com/binomial-distribution-formula/
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