MAULANA AZAD NATIONAL INSTITUTE OF TECHNOLOGY, BHOPAL

REPORT ON

BINOMIAL PROBABILITY METHOD

Subject:Statistics & Demography [UD 122 ]

2013Submitted to:Dr. C.K VermaDepartment of Mathematics (Applied Mathematics)

Submitted by:Shailendra Kumar

II sem Sch. No. 122110105

Department of Architecture and

PlanningMANIT Bhopal

Contents:

1. Introduction:1.1. What is statistics?1.2. Why the need for statistics?1.3. Terminology:

2. Probability tests3. Laws of probability4. Binomial probability method5. General formulae for finding binomial probability6. Mean, Variance, and Standard Deviation of binomial

distribution7. Problems on binomial probability distribution8. Bibliography

1.Introduction:1.1. What is statistics? A branch of mathematics that provides techniques to analyse

whether or not your data is significant (meaningful). Statistical applications are based on probability statements. Nothing is “proved” with statistics. Statistics are reported. Statistics report the probability that similar results would occur if

you repeated the experiment.

1.2. Why the need for statistics? Statistics are used to describe sample populations as estimators of

the corresponding population. Many times, finding complete information about a population is

costly and time consuming. We can use samples to represent a

population.

1.3. Terminology: Variable: It is a continuous data. Data values can be any real

number or Measured data. Continuous variables: type of numbers associated with measuring

or weighing; any value in a continuous interval of measurement. Examples: Weight of students, height of plants, time to flowering

Discrete variables: type of numbers that are counted or categorical. Examples: Numbers of boys, girls, insects, plants

Population: It includes all members of a group. Example: all 9th grade students in America, Number of 9th grade students at SR, No absolute number

Sample: It is used to make inferences about large populations. Samples are a selection of the population. Example: 6th period Accelerated Biology

Probability : It means likelihood. It is a measure or estimation of how likely it is that something will happen or that a statement is true.

Parameters: Quantities that describe a population characteristic. They are usually unknown and we wish to make statistical inferences about parameters. Different to perimeters.

Descriptive Statistics: Quantities and techniques used to describe a sample characteristic or illustrate the sample data e.g. mean, standard deviation, box-plot

2.Probability tests What to do when you are comparing two samples to each other and

you want to know if there is a significant difference between both sample populations

(example the control and the experimental setup) How do you know there is a difference How large is a “difference”? How do you know the “difference” was caused by a treatment and

not due to “normal” sampling variation or sampling bias?

3.Laws of probability: The results of one trial of a chance event do not affect the results of

later trials of the same event. p = 0.5 ( a coin always has a 50:50 chance of coming up heads)

The chance that two or more independent events will occur together is the product of their changes of occurring separately. ( one outcome has nothing to do with the other)

The probability that either of two or more mutually exclusive events will occur is the sum of their probabilities (only one can happen at a time).

4.Binomial probability method

The binomial probability method offers a simple but very useful model.

A binomial method is characterized by trials which either end in success (heads) or failure (tails). These are sometimes called Bernoulli trials .

Suppose we have n Bernoulli trials and p is the probability of success on a trial. Then this is a binomial model if

1. The Bernoulli trials are independent of one another.2. The probability of success, p, remains the same from trial to trial.

A binomial method is an experiment which satisfies these four conditions:

1. A fixed number of trials2. Each trial is independent of the others3. There are only two outcomes4. The probability of each outcome remains constant from trial to

trial.

Examples of binomial experiments

Tossing a coin 20 times to see how many tails occur. Asking 200 people if they watch ABC news. Rolling a die to see if a 5 appears.

Examples which aren't binomial experiments

Rolling a die until a 6 appears (not a fixed number of trials) Asking 20 people how old they are (not two outcomes) Drawing 5 cards from a deck for a poker hand (done without

replacement, so not independent)

5.General formulae for finding binomial probability

6.Mean, Variance, and Standard Deviation of binomial

rnrr

n qpcrP )(

r =successes out of n trials

p = probability of success

q=1-p = probability of failure

n C r=n!(n−r ) !r !

n !=nx (n−1 )x (n−2) x . . .. 1

The mean, variance, and standard deviation of a binomial distribution are extremely easy to find.

7.Problems on binomial probability distribution

Q1. A coin is tossed four times, what is the probability of getting.

1. No head2. Exactly 1 head3. Exactly 2 heads4. At least 2 heads5. At most 2 heads

SOL (i): By formula Here,r=0p= probability of success=1/2q= probability of failure=1/2

npq

npq

np

2

Probability

n = number of trials

rnrr qpCrP n )(

P(0 )=4 C0(12

)0(12

)4−0

=4 !(4−0 )! 0!

(12

)0(12

)4−0

¿116

SOL (ii): By formulaHere,r=1p= probability of success=1/2q= probability of failure=1/2

SOL (iii): By formula Here,r=2p= probability of success=1/2q= probability of failure=1/2

rnrr qpCrP n )(

rnrr qpCrP n )(

P(0 )=4 C0(12

)0(12

)4−0

=4 !(4−0 )! 0!

(12

)0(12

)4−0

¿116

P(1)=4 C1(12

)1(12

)4−1

=4 !(4−1 )!1 !

(12

)1(12

)3

¿4 x3 !3 !

(116

)

¿14

P(2 )=4 C2(12

)2(12

)4−2

=4 !(4−2 )!2 !

(12

)2 (12

)2

¿4 x3 x2 !2 ! x2 !

(116

)

¿38

SOL (iv):

By formula Here,r=2+3+4p= probability of success=1/2q= probability of failure=1/2

SOL (v):

By formula rnrr qpCrP n )(

rnrr qpCrP n )(

P(2 )=4 C2(12

)2(12

)4−2

=4 !(4−2 )!2 !

(12

)2 (12

)2

¿4 x3 x2 !2 ! x2 !

(116

)

¿38

P(2 )+P (3 )+P(4 )=38

+4 C3(12

)3(12

)4−3+ 4 C4(12

)4(12

)4−4

=38

+4 !( 4−3)!3 !

(12

)3(12

)1+4 !(4−4 )! 4 !

(12

)4 (12

)0

¿38

+4 x 3!3!

(116

)+4 !4 !

(116

)

¿1116

Here,r=0+1+2 or 1-(iv)p= probability of success=1/2q= probability of failure=1/2

Q2. The mean and the variance of binomial distribution are 4 and 4/3.

i. Find the probability of two successii. The probability of more than two success

SOL:

Mean of binomial distribution=np=4Variance of binomial distribution=npq=4/3Putting the value of np in variance we get4xq=4/3Then q=1/3So, p=1-qi.e p=1-1/3Therefore, p=2/3And n=4/3pqi.e n=6

(i)By formula rnr

r qpCrP n )(

P(0 )+P(1)+P(2 )=1−P( iv )

=1−(1116

)

=1516

Here, r=2,p= probability of success=2/3,q= probability of failure=1/3

SOL (ii) By formulaHere, r=3+4+5+6p= probability of success=2/3,q= probability of failure=1/3

Q3. A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions.

rnrr qpCrP n )(

P(2 )=6 C2 (23

)2 (13

)6−2

¿6 !(6−2 )!2 !

(49

)(181

)

¿6 x5 x 4 !4 ! x2 !

(4729

)

¿20243

P(3 )+P (4 )+P(5 )+P (6)=6 C3 (23

)3 (13

)6−3+6 C4(23

)4 (13

)6−4+6 C5(23

)5 (13

)6−5+6 C6 (23

)6 (13

)6−6

¿6 !(6−3 )!3 !

(827

)(127

)+6 !(6−4 )!4 !

(1681

)(19

)+6 !(6−5) !5 !

(32243

)(13

)+6 !(6−6 )! 6!

(64729

)

¿592729

What is the probability of getting exactly 6 questions correct on this test?

SOL: By formulaHere,n=10r=6p= probability of success=1/5=0.2,q= probability of failure=4/5=0.8

Q4. When rolling a die 100 times, what is the probability of rolling a "4" exactly25 times?

SOL: By formulaHere,n=100r=25p= probability of success=1/6q= probability of failure=5/6

rnrr qpCrP n )(

rnrr qpCrP n )(

P(6 )=10 C6 (0. 2 )6(0 . 8)10−6

¿10 !(10−6 )!6 !

(0 . 000064 )(0 . 4096)

¿0 .005505024≈0 . 006

P(25 )=100 C25(16

)25(56

)100−25

¿100 !(100−25 )!25 !

(16

)25(56

)75

¿0 .0098258819≈0.010

Q5. At a certain intersection, the light for eastbound traffic is red for 15 seconds, yellow for 5 seconds, and green for 30 seconds. Find the probability that out of the next eight eastbound cars that arrive randomly at the light, exactly three will be stopped by a red light.

SOL:

By formulaHere,n=8r=3p= probability of success=15/50q= probability of failure=35/50

Q6. The manufacturing sector contributes 17% of Canada's gross domestic product. A customer orders 50 components

rnrr qpCrP n )(

P(25 )=100 C25(16

)25(56

)100−25

¿100 !(100−25 )!25 !

(16

)25(56

)75

¿0 .0098258819≈0.010

P(3 )=8 C3 (1550

)3(3550

)8−3

¿8 !(8−3 )!3 !

(1550

)3 (3550

)5

¿0 .25412184≈0 .254

from a factory that has 99% quality production rate (99% of the products are defect-free). Find the probability that:

i. none of the components in the order are defective.ii. there are at least two defective products in the order.

SOL (i):

By formula Here,n=50r=50p= probability of success=0.99q= probability of failure=0.01

SOL(ii):

By formula Here,n=50r=0+1+2+3…+48p= probability of success=0.99q= probability of failure=0.01

rnrr qpCrP n )(

rnrr qpCrP n )(

P(50 )=50 C50( 0. 99 )50(0. 01 )50−50

¿50 !(50−50 )!50 !

(0. 99 )50(0. 01 )0

¿0 .605

P(2 failure )=p (0 success )+.. . ..+ p( 48 success )=1−p (49 success)−p (50 success )=1−50 C49(0. 99 )49(0 . 01)1−50 C50(0 .99 )50(0 . 01)0

¿1−0. 306−0 .605¿0 .089

Q7. A baseball player comes to bat 4 times in a game. The chance of a strike-out for this player is 30%. Find all possible outcomes and their probabilities.

SOL(i):

By formula Here,n=4(number times to bat)r=0, 1, 2, 3,4p= probability of success=0.3q= probability of failure=0.7

rnrr qpCrP n )(

P(2 failure )=p (0 success )+.. . ..+ p( 48 success )=1−p (49 success)−p (50 success )=1−50 C49(0. 99 )49(0 . 01)1−50 C50(0 .99 )50(0 . 01)0

¿1−0. 306−0 .605¿0 .089

Q8. A survey indicates that 41% of American women consider reading as their favourite leisure time activity. You randomly select four women and ask them if reading is their favourite leisure-time activity. Find the probability that i. exactly two of them respond yes

ii. at least two of them respond yesiii. Fewer than two of them respond yes. SOL (i): By formula Here,n=4r=2p= probability of success=0.49q= probability of failure=0.51

SOL (ii):

By formula Here,n=4r=2,3,4p= probability of success=0.49

rnrr qpCrP n )(

rnrr qpCrP n )(

P(2 )=4 C2(0 .41 )2(0 .59 )4−2

¿4 !( 4−2)!2!

(0 . 41)2 (0 .59 )4−2

¿244

( .1681 )( .3481 )

¿6( . 1681)( . 3481)=.35109366

q= probability of failure=0.51

SOL (iii):

By formula Here,n=4r=0, 1p= probability of success=0.49q= probability of failure=0.51

rnrr qpCrP n )(

P(2 )=4 C2(0 .41 )2(0 .59 )4−2=.351093

P(3 )=4 C3( 0.41 )3(0 .59)4−3=0 .162653

P( 4 )=4 C4 (0 .41)4 (0 . 59)4−4=0 .028258

P( x≥2 )=P(2)+P(3 )+P(4 )=. 351093+. 162653+028258¿0 .542

P(0 )=4 C0 (0 . 41)0 (0.59 )4−0=0 .121174

P(1)=4 C1(0 .41)1 (0 .59 )4−1=0. 336822

P( x<2)=P(0)+P(1 )=. . 121174+ .336822¿0 .458

Q9. 65% of American households subscribe to cable TV. You randomly select six households and ask each if they subscribe to cable TV. Construct a probability distribution for the random variable, x. Then graph the distribution.

SOL:

By formula Here,n=6r=0, 1, 2 , 3, 4, 5, 6p= probability of success=0.65q= probability of failure=0.35

x 0 1 2 3 4 5 6

P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075

rnrr qpCrP n )(

P(0 )=6 C0(0 .65)0(0 .35)6−0=0 .002

P(1)=6 C1 (0. 65 )1(0 . 35)6−1=0.020

P(2 )=6 C2 (0 .65 )2(0 . 35 )6−2=0 . 095

P(3 )=6 C3(0 .65)3 (0 .35 )6−3=0 . 235

P( 4 )=6 C4( 0. 65 )4 (0 .35 )6−4=0 .328

P(5 )=6 C5 (0 .65)5 (0. 35 )6−5=0 . 244

P(6 )=6 C6(0 .65 )6(0 .35)6−6=0 .075

Q10. A six sided die is rolled 3 times. Find the probability of rolling exactly one 6.

SOL:

By formula Here,n=3r=1p= probability of success=1/6q= probability of failure=5/6

0 1 2 3 4 5 60

0.050.10.150.20.250.30.35

P(x)

rnrr qpCrP n )(

P(1)=3 C1 p1q3−1

¿3 !(3−1)!1!

(16

)1(56

)3−1

¿3(16

)(56

)2

¿3(16

)(2536

)

¿2572

≈0 . 347

8.Bibliography

1. https://www.khanacademy.org/math/probability/independent-dependent-probability

2. https://en.wikipedia.org/wiki/Probability3. http://people.richland.edu/james/lecture/m170/ch06-bin.html4. http://www.mathwords.com/b/binomial_probability_formula.htm5. http://www.stats.gla.ac.uk/steps/glossary/

probability_distributions.html#binodistn6. http://www.stat.wmich.edu/s160/book/node33.html