Numerous domains, including economics, engineering, physics, and biology, utilise the binomial distribution as a potent statistical tool. It depicts the conclusion of a straightforward experiment with only two possible outcomes, success and failure, and a constant chance of success.
We will examine the binomial distribution in depth, including its definition, formula, properties, applications, and examples.
Table of Contents
What exactly is the Binomial Distribution?
Formula for Binomial Distribution
Attributes of the Binomial Distribution
Uses of the Binomial Distribution
Illustrations of Binomial Distribution
Conclusion
What exactly is the Binomial Distribution?
Binomial distribution is a probability distribution that defines the number of successes in a certain number of independent trials where only two outcomes, success or failure, are possible for each trial. Each trial has a consistent probability of success and probability of failure.
For instance, if we flip a coin five times and wish to determine the likelihood of obtaining exactly three heads, we might utilise the binomial distribution.
The binomial distribution is named after the Swiss mathematician Jacob Bernoulli, who established it in his 1713 work Ars Conjectanda.
Formula for Binomial Distribution
The likelihood of obtaining exactly k successes in n independent trials of a binomial experiment with probability p of success on a single trial can be calculated using the following formula.
P(k) = (n C k) * p^k * (1-p)^(n-k)
where
(n pick k) = n!/k!(n-k)! reflects the number of methods to select k successes out of n trials.
pk reflects the likelihood of achieving k achievements.
(1-p)(n-k) reflects the likelihood that n-k failures will occur.
“!” represents the factorial operation, where n! = n*(n-1)(n-2)…*1.
Attributes of the Binomial Distribution
The binomial distribution contains several essential characteristics, including:
a. The formula for the mean of the binomial distribution is:
mean = np,
where p is the probability of success on a single trial and n is the number of trials.
b. The variance of the binomial distribution can be calculated using
variance = np(1-p)
c. The binomial distribution has a probability mass function and is discrete.
d. When p = 0.5, the binomial distribution is symmetric.
e. When n is big, the binomial distribution approaches the normal distribution.
Uses of the Binomial Distribution
The binomial distribution has numerous applications in the real world, including:
a. In manufacturing, binomial distribution is used to determine the number of defective goods in a sample for quality control purposes.
b. In genetics, the binomial distribution is used to model the inheritance probability of a genetic trait.
c. Medicine: The binomial distribution is used to calculate the likelihood that a medicine will be effective for a given proportion of patients.
d. Elections: The binomial distribution is used to estimate the likelihood that a candidate will win an election.
e. The binomial distribution is used to estimate the likelihood of a team winning a game.
Illustrations of Binomial Distribution
Example 1: Ten times a coin is flipped. How likely is it to receive exactly five heads?
This issue can be resolved by applying the binomial distribution formula.
n = 10, k = 5, p = 0.5
P(5) equals (10 choose 5) * (0.5)5 * (1-0.5)(10-5)
P(5) = 252 * 0.5^5 * 0.5^5
P(5) = 0.246
Therefore, the probability of obtaining exactly five heads from 10 tosses of a coin is 0.246%.
A basketball player has a sixty percent chance of making a free shot. What is the likelihood that he will make 15 of 20 free shots if he attempts 20?
Solution: n = 20, k = 15, p = 0.6
P(15) equals (20 choose 15) * (0.6)15 * (1-0.6)(20-15)
P(15) = 15,504 * 0.6^15 * 0.4^5
P(15) = 0.139
The probability of making exactly 15 of 20 free throws is therefore 0.139.
Conclusion
Binomial distribution is a strong tool that enables us to compute the probability of a fixed number of successes or failures in a fixed number of trials with just two possible outcomes for each trial. Numerous sectors, including engineering, economics, physics, and biology, can utilise it. Understanding the binomial distribution is crucial for anyone working with probability and statistics.
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