Note on Binomial Distribution Examples

Binomial Distribution works on the two parameters, n and p. n is the number of trials and p is the probability of one outcome. 

Let’s say, you have tossed a coin five times, and the probability of getting a tail is ½. So, on the scale of the binomial distribution, n will be equal to 5 ( n = 5 ), i.e., the number of times, the coin has been tossed and p will be equal to ½, the probability of any one outcome ( p = ½ ). At a glance, this is the Binomial Distribution short overview. Now in this article, we will focus on some examples of Binomial Distribution.

Some Basic Definitions of Binomial Distribution

Bernoulli Trials: We will be getting only two outcomes either success or failure in an experiment. These single outcome results are known as Bernoulli Trials.

Bernoulli Distribution: Bernoulli distribution is preferred for representing a single condition or experiment, where n =1.

Bernoulli Process:  When we are having more than two outcomes, that is a series of results, then this sequencing is known as the Bernoulli Process.

Let us understand some examples of Binomial Distribution

1. To find the number of used and unused materials while manufacturing. 

Example -1: A bike manufacturing company has two plants A and B. Plant A manufactures 75% of bikes and plant B manufactures 25%. 60% of the bikes at plant A and,70% of the bikes at plant B are of Standard Quality. A bike is chosen at random and is found to be of Standard quality. Then find the probability for the chosen bike from Plant A. 

Ans : Let, E be the event that the bike is selected of standard quality. 

A1 is the event that the Bike is manufactured in plant A.

B1 is the event that the bike is manufactured in plant B. 

Hence, 

P(A1) = 75/100 = ¾

P(B1) = 25/100 = ¼

P(E|A1) = Probability of event that says the bike is manufactured in plant A.

P(E|B1) = Probability of event that says the bike is manufactured in plant B.

P(E|A1) = 60 / 100 = 6/10

P(E|B1) = 70 / 100 = 7/10

P(A1|E) = probability of the event that a standard quality bike is manufactured in plant A. 

P(A1|E) =                    P (A1) × P (E |A1) 

                P (A1) . P (E | A1 ) + P (B1) . P (E |B1 )

=      ¾ × 6/10

¾ . 6/10 + ¼ . 7/10

= 9/20 × 8/5

=  18/25

1. Examples on voting problems

1. In a city, there are total 10 voters. Then find the probability that 4 to 6 of them will vote ?

Ans:.                     P(4 ≤ X ≤ 6) = P(X = 4) + P(X = 5) + P(X = 6)

    = 10 4 0.5540.456 + 10 5 0.555 0.455 + 10 4 0.556 0.455

    ≈0.160+0.234+0.238

    = 0.632

1. The number of men and women working in a company

Question: In a locality, 60% of people are working in IT companies. If 10 of them are selected randomly, then find the probability that exactly 7 of them are men.

Answer: Step – 1: Identify n  and X for the given statement. 

n = number of randomly selected items.

Here, n = 10 ( No. of randomly selected persons )

X = number of persons who asked to be selected 

Here, X = 7 (No. of 7 men who asked to be selected)

Step – 2: We have the formula, 

= n!n-X! X!

= 10!10-7! 7!

= 120

Step – 3: Probability of success (p);

Given, p = 60% = 0.6

  Probability of Failure (q);

q = 1 – p

q = 1 – 0.6 = 0.4

Step – 4: Find, pX = (0.6)7

    = 0.0279

Step – 5: Find, q(n – X) = 0.4(10 – 7)

= 0.43

= 0.064

Step – 6: Probability;

P(7 of 10 are men) =  n!n-X! X! . pX . q(n – X)

        = 120 × 0.0279 × 0.064

        = 0.215

Conclusion

In this article, we have read about binomial distribution. We have discussed the basic conditions for Binomial distribution according to which, the binomial distribution number of trials is fixed. And binomial trials must be independent trials. Binomial distributions are discrete. You can understand the binomial distribution, we have some examples which can make you better understand it. These examples are, Tossing a Coin, Rolling a dice, probability for writing an examination, counting the number of votes, etc.