Binomial Distribution

This is a discrete random variable, where the process of obtaining the Binomial distribution is called “Bernoulli “ process. An experiment that often consists of repeated trials, each with two possible outcomes, which could be labeled as “success” or “failure”. This experiment is known as binomial experiment.

A binomial experiment is one that possesses the following properties:

1. The experiment consists of n repeated trials.

2. Each trial has only 2 possible outcomes that can be classified as “Success” or “Failure”.

3. The probability of a success and failure , denoted by pand q, remains constant from trial to trial. 4. The repeated, trials are independent.

Formula for success in the Bernoulli process.

The probability of r success from n trial is :

[pic]

where; r - the number of success n – the number of trial p – the probability of success from one trial and q – the probability of failure from one trial. q = 1 - p [pic] with mean, [pic] and variance, [pic]

It is written as X ~ B(n,p) and read as X is Binomial distribution with parameters n success and the probability of success, p.

Example 7.4

One coin is tossed 5 times. If X is random variable for number of heads . Find the probability of getting
(a) no heads
(b) one head
(c) 3 heads
(d) at least 3 heads

Solution
X is the random variable represents the number of heads.
The possible outcome of one trial ( i.e one tossed ) is a Head or a Tail.
Thus S = {H, T}, [pic] p is the probability of getting Head, [pic] q is the probability of not getting Head, [pic] n is number of trial ( i.e number of tossed ), n = 5

Thus, X~ B(5, ½)
As the number of trial, n = 5, we can obtain either 0, 1, 2, 3, 4 or 5 heads from the trials. Thus we can take the value of r as 1, 2, 3, 4 and 5.
|r |P ( X = r ) |
|0 |[pic] |
|1 |[pic] |
|2 |[pic] |
|3 |[pic] |
|4 |[pic] |
|5 |[pic] |

(a) P ( no heads ) = [pic]

(b) P ( one head ) = [pic]

a) P ( 3 heads ) = [pic]

b) P ( at least 3 heads ) = [pic]

Example 7.5

In a process of producing screws, 10% of the screw was rejected because the screws are too soft. What is the probability for a sample of 12 screws that contains
(a) 2 rejected screws
(b) not more than 2 rejected screws.

Solution
X is the random variable of the number of rejected screws p - probability of getting rejected screws. P = [pic] n – represent number of screws in the sample, n = 12 r - represent the number of screw which would be rejected. X ~ B (12, 0.1)

(a) [pic]P( X =2)

(b) [pic] [pic] = 0.8891

Example 7.6
A box contains seeds where 1% of them are cannot grow. If the box contains 10,000 seeds, determine the mean and the variance of the seeds which cannot grow.

Solution X is the random variable number of seeds which cannot grow. p – probability of a seeds which cannot grow.[pic] n – is the number of the seeds in the box, n = 10,000.

X~ B (10,000, 0.01) Mean, [pic]seeds Variance, [pic]seeds.

Exercise 7.2

1. If X represents the no. of heads from 5 toss of a coin. a) Find the value of X. b) Calculate the value of P(X). c) What is the probability for at least 3 heads in a try?

2. A die is thrown 5 times. Calculate the probability when the lands at 4, if a) twice b) 3 times.

3. 25% of a local university who registered for the first year needs additional class for mathematics. If 6 students are chosen at random, find the probability a) 1 student needs the additional class b) 2 students need the additional class c) 3 students who need the additional class.

4. If X represents the number of broken pencils from 5 pencils chosen at random in a box of 100 pencils, where 10 are broken. Find the probability a) P(X = 3) b) P(X ≥ 3) c) P(X ≤ 2)

5. In an examination of 10 objective questions, every one contains 5 answers but only one that is correct. If one student that has never study, sit for examination could only answer by guessing the right answer. Find the probability a) none of the answer are correct b) only 3 answer are correct.

6. Probability of a man between 20-24 years married is 0.2. 20 men are chosen from the age group, find a) the probability that 9 has gotten married b) the probability that less than 3 are already married c) µ , the number of men from the group who has already married.

7. 40% from Kedah’s population visit Langkawi Island every year. 50 people are chosen from the population, find the probability a) at least 30 had visited Langkawi b) at least 15 had visited Langkawi.

8. It was found out that 75% in a village had never read a book. 20 villagers were chosen, find the a) 5 had never read a book b) more than 12 had never read a book c) less than 8 had never read a book.

9. 20% of the nails made by a machine are defectives. 10 nails were taken, find the probability a) 3 nails are defective b) at least 6 are defective c) less than 4 are defective.

10. Find the probability when n = 7 and p = 0.2, in a Binomial Distribution. a) p(r =5) (c) p(r > 2) b) p(r < 8) (d) p(r ≥ 4)

11. Find the mean and the standard deviation for the following binomial distribution: a) B(15, 0.20) b) B(8, 0.42) c) B(72, 0.06) d) B(29, 0.49) e) B(642, 0.2)