Key Properties and Real-Life Applications of Binomial Distribution
Bernoulli trial, binomial distribution and Bernoulli distribution are briefly explained in this article. Let us first learn about Bernoulli trials. Bernoulli trials are also known as binomial trials as there are only possible outcomes in Bernoulli trials i.e success and failure whereas in a binomial distribution, we get a number of successes in a series of independent experiments. A Bernoulli distribution is the probability distribution for a series of Bernoulli trials where there are only two possible outcomes. In this article, we will discuss,bernoulli trial binomial distribution, bernoulli trial formula, bernoulli trial example, bernoulli distribution, bernoulli distribution examples, properties of bernoulli distribution, how bernoulli trial is related to binomial distribution etc.
Bernoulli Trials
In the field of probability, the experimentation of different concepts led to major mathematical theories. Let us assume one experiment which will be finite in number. It should have only two results as outcome one will be termed as success and the other one as a failure. And in all the experiment terms the probability of events failure and success does not change. This setup of the experiment is known as the Bernoulli trial. This was created by Jacob Bernoulli, a Swiss mathematician and published in his book Ars Conjectandi in 1713. This helped us in understanding the nature of probability better.
Definition
A successive event in a sequence of independent experiments where there are only two possible outcomes and the probability of the outcomes remains the same in each event. When we conduct these events in a succession for a finite number of times then the series of experiments is called Bernoulli's trials.
Binomial Distribution
The binomial distribution is a graphical representation of the results of Bernoulli's trials. It gives us the idea of the probability of events throughout the experiment successions. This can also be said as the frequency distribution of the probability of a given number of successes in a random number of repeated independent Bernoulli trials. If Bernoulli's trials are the experiment then the binomial theorem can be said as the result or finding of the experiment.
The Use
Say we have to find the probability of heads in a coin toss then we can easily find it by using success occurrence divided by sample space formula and the same can be said for finding the probability of two heads in two consecutive coin tosses. If we have to find the probability of two heads and three tails in 5 consecutive coin tosses then it will be difficult but solvable by the above method but the complexity is very huge. But say we have to find the probability of even no heads counting in 20 consecutive coin tosses. It will be next to impossible for us to fund the probability by the above method. In these complex cases, we will have to use the binomial distribution formula which will help us to find the probability of complex predictable problems.
Formula
\[P_{r} = (\frac{n}{r})p^{r} q^{n-r}\]
n= number of trials
r= number of success
p= probability of success
q=probability of failure
More About Binomial Distribution
The binomial distribution is a kind of probability distribution that has two possible outcomes. In probability theory, binomial distributions come with two parameters such as n and p.
The probability distribution becomes a binomial probability distribution when it satisfies the below criteria.
The number of trials must be fixed.
The trials are independent of each other.
The success of probability remains similar for every trial.
Each trial has only two outcomes namely success or failure.
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Bernoulli Trial
A random experiment that has only two mutually exclusive outcomes such as “success “and “not success “is known as a Dichotomous experiment.
If a Dichotomous experiment is repeated many times and if in each trial you find the probability of success p (0< p <1) is constant, then all such trials are known Bernoulli trials.
As, Bernoulli trials has only two possible outcomes, it can easily frame as “yes” or “no” questions
Bernoulli Trials Example
The Bernoulli trial example will explain the concept of bernoulli trial in two different situation:
8 balls are drawn randomly including 10 white balls and 10 black balls. Examine whether the trials are Bernoulli trials if the balls are replaced and not replaced.
In the first trial, when the ball is drawn with replacement, the probability of success (say, the black ball) is 10/20 = ½ which is similar for all 8 trials. Hence, the trials including the drawing of balls with replacement are considered as Bernoulli trials.
In the second trial, when drawn without replacement, the probability of success (say, the black balls) changes with the number of trials =10/20 = ½ for second trials, the probability of success p =9/19 which is not similar to the first trial. Hence, the trails including the drawing of balls without replacement are not considered as Bernoulli trial
Bernoulli Trials Conditions
The Bernoulli trial has only two possible outcomes i.e. success or failure.
The probability of success and failure remain the same throughout the trials.
The Bernoulli trials are independent of each other.
The number of trials is fixed.
If x is the probability of success then the probability of failure is 1-x.
Bernoulli Trials Formula
Here, you can see the Bernoulli trial formula in Bernoulli Math.
Let us take an example where n bernoulli trials are made then the probability of getting r successes in n trials can be derived by the below- given bernoulli trials formula.
P(r) = Cn pr qn-r
The term n! / r!(n!-r!)! is known as a binomial coefficient.
Bernoulli Trials and Binomial Distribution
The student will be able to design a Bernoulli trial or experiment
The student will be easily able to use binomial formula
The student will be able to design a binomial distributions
The students will be able to compute applications including Bernoulli trials and binomial distribution’
Bernoulli Distribution
A Bernoulli distribution in Bernoulli Maths is the probability distribution for a series of Bernoulli trials where there are only two possible outcomes. It is a kind of discrete probability distribution where only specific values are possible. In such a case, only two values are possible;e ( n=0 for failure and n=1 for success). This makes the Bernoulli distribution the simplest form of the probability distribution that persists.
Bernoulli Distribution Examples
Some of the bernoulli distribution examples given in bernoulli Maths are stated below:
A newly born child is either a girl or a boy ( Here, the probability of a child being a boy is roughly 0.5)
The student is either pass or fail in an exam
A tennis player either wins or losses a match
Flipping of a coin is either a head or a tail.
Properties of Bernoulli Distribution
Here, you can find some of the properties of bernoulli distribution in bernoulli Maths.
The expected value of the bernoulli distribution is given below.
E(X) = 0 * (1-P) + 1 * p = p
The variance of the bernoulli distribution is computed as
Var (X) = E(X²) -E(X²) = 1² * p +0² * ( 1-p) - p² = p - p² = p (1-p)
The mode, the value with the highest probability of appearing, of a Bernoulli distribution is 1 if p > 0.5 and 0 if < 0.5, success and failure are equally likely and both 0 and 1 are considered as modes.
The basic properties of Bernoulli Distribution can be computed by considering n=1 in probability
Quiz Time
1. How many possible outcomes can there be for Bernoulli trials?
More than 1
More than 2
Exactly 1
Exactly 2
2. What will be the variance of the Bernoulli trials, if the probability of success of the Bernoulli trial is 0.3.
0.3
0.7
0.21
0.09
3. The mean and variance are equal in binomial distributions.
True
False
Solved Examples
1. If the probability of the bulb being defective is 0.8, then find the probability of the bulb not being defective.
Solution:
Probability of bulb being faulty, p = 0.8
Probability of bulb not being defective, q = 1-p = 1-0.8= 0.2
Hence, probability of bulb not being defective, q = 0.2
2. In an exam, 10 multiple choice questions are asked where only one out of four questions are correct. Find the probability of getting 5 out of 10 questions correct in an answer sheet.
Solution: Probability of getting an answer correct, p = ¼
Probability of getting an answer incorrect , q = 1-p = 1
Probability of getting 5 answers correct, P(X=5) = (0.25)5 ( 0.75)5 = 0.5839920044
3. Toss a coin 12 times. What is the probability of getting 7 heads?
Solutions:
Number of trials(N) = 12
Number of success (r)= 7
Probability of single trial (P) = ½ = 0.5
nCr =
n!r!
n!r! * (n-r)!
12!7! (12-7)!
= 95040120
= 792
pr= 0.57=
0.0078125
To find (1-p)n-r,calculate (1-p) and (n-r)
Solving P (X=r) = nCr .pr. (1-p)n-r
= 792 * 0.0078125 *0.03125
= 0.193359375
The probability of getting 7 heads is 0.19
FAQs on Bernoulli Trials and Binomial Distribution Explained
1. What is a Bernoulli trial in probability?
A Bernoulli trial is a random experiment that has exactly two possible outcomes: 'success' or 'failure'. A key characteristic is that the probability of success, denoted by 'p', remains constant every time the experiment is repeated. For example, a single toss of a fair coin is a Bernoulli trial, where getting a 'Head' can be considered a success.
2. What are the key conditions that define a Bernoulli trial?
For an experiment to be classified as a Bernoulli trial, it must satisfy two main conditions:
Two Outcomes: The trial must have only two mutually exclusive outcomes, typically labelled as success and failure.
Independence and Constant Probability: Each trial must be independent of the others. This means the outcome of one trial does not affect the outcome of another, and the probability of success (p) is the same for every trial.
3. What is the main difference between a Bernoulli trial and a Binomial Distribution?
The main difference lies in the number of experiments performed. A Bernoulli trial refers to a single experiment with two outcomes (e.g., one coin toss). In contrast, a Binomial Distribution describes the probability of observing a specific number of successes in a fixed number of independent Bernoulli trials (e.g., the probability of getting 7 heads in 10 coin tosses).
4. What is the formula for Binomial Distribution, and what do its components represent?
The Binomial Distribution formula calculates the probability of getting exactly 'k' successes in 'n' trials. The formula is: P(X=k) = C(n, k) * pk * q(n-k). Here's what each component means:
n: The total number of independent trials.
k: The specific number of successes you want to find the probability for.
p: The probability of success in a single trial.
q: The probability of failure in a single trial, which is always (1-p).
C(n, k): The number of combinations for choosing k successes from n trials, calculated as n! / (k!(n-k)!).
5. Can you provide a real-world example of Bernoulli trials and a Binomial Distribution?
Certainly. Imagine a quality control process where a new light bulb is tested. The bulb either works (success) or is defective (failure). Testing a single bulb is a Bernoulli trial. If you test a box of 20 bulbs and want to find the probability that exactly 2 are defective, you would use the Binomial Distribution, where n=20 and k=2.
6. How does a sequence of independent Bernoulli trials lead to a Binomial Distribution?
A Binomial Distribution is fundamentally built upon Bernoulli trials. When you conduct a single experiment with two outcomes, it's a Bernoulli trial. When you decide to repeat this exact experiment a fixed number of times (e.g., 'n' times) and each trial is independent, the collection of these trials is called a Bernoulli process. The Binomial Distribution is the mathematical model that provides the probabilities for the various possible counts of 'successes' across that entire sequence of trials.
7. Why is drawing cards from a standard deck without replacement NOT considered a series of Bernoulli trials?
This scenario violates the core condition of independence and constant probability. For the first draw, the probability of picking a King is 4/52. If you draw a King and do not replace it, the probability of drawing another King on the second trial changes to 3/51. Since the outcome of the first trial affects the probabilities of subsequent trials, the trials are not independent, and therefore, this cannot be modelled as a series of Bernoulli trials.
8. What is the importance of the parameters 'n' and 'p' in a Binomial Distribution?
The parameters 'n' (number of trials) and 'p' (probability of success) are crucial as they completely define a Binomial Distribution. The parameter 'n' determines the number of possible outcomes (from 0 to n successes), while 'p' determines the shape and skewness of the distribution. A 'p' value of 0.5 results in a symmetric distribution, whereas values closer to 0 or 1 make it skewed.
9. How do you calculate the mean and variance for a Binomial Distribution as per the CBSE syllabus for 2025-26?
For a Binomial Distribution defined by parameters n and p, the mean and variance are calculated using simple formulas:
Mean (μ): The expected number of successes is calculated as μ = np.
Variance (σ²): The measure of the spread or variability of the distribution is calculated as σ² = npq, where q = 1-p.
These formulas are essential for understanding the central tendency and dispersion of the outcomes in a binomial experiment.
10. When would a Binomial Distribution be an inappropriate model for an experiment with two outcomes?
A Binomial Distribution is inappropriate even with two outcomes if its core assumptions are not met. The most common reasons include:
Lack of Independence: If the trials affect each other, such as sampling without replacement from a small population.
Changing Probability: If the probability of success 'p' does not remain constant for all trials.
Non-fixed Number of Trials: The Binomial model requires 'n' to be fixed before the experiment begins. If you continue trials until you achieve a certain number of successes, you would need a different model, like the Negative Binomial Distribution.
