# I. Determine whether each of t

I. Determine whether each of the following random variables hasa binomial distribution. If it does, identify the values of theparameters n and p. If not, explain.

a) X is the number of times five is rolled in 15 rolls of afair, six-sided die.

b) X is the number of multiple-choice questions a student getsright on a 30-question test, when each question has four choicesand the student is completely guessing on each question.

c) X is the same as (b), but the first 15 questions have fourchoices and the last 15 have three answer choices.

d) X is the number of women in a random sample of 12 students,from a class comprising 25 women and 20 men.

e) X is the total weight of 16 randomly selected oranges

f) X is the number of oranges, out of a random sample of 16,that weight more than 170 grams

Answer:

Binomial Distribution

If ** ‘X’** is the randomvariable representing the number of successes, the probability ofgetting

**successes and**

*‘r’***failures, in**

*‘n-r’***trails,**

*‘n’***probability of success**

*‘p’***is givenby the probability function**

*‘q’=(1-p)*a.X is the number of times five is rolled in 15 rolls of a fair,six-sided die.

Success : Five is rolled

X :Number of successes: Number of times five is rolled:

n : Number of trails : 15 rolls of a fair die

p: Probability of success : Probability of rolling a five=1/6

Hence ‘X’ has a binomial distribution with n=15 and p=1/6

b. X is the number of multiple-choice questions a student getsright on a 30-question test, when each question has four choicesand the student is completely guessing on each question.

Success : questions a student gets right

X :Number of successes: number of multiple-choice questions astudent gets right

n : Number of trails : 30-question test

p: Probability of success : Probability of getting a questionright = 1/4 (As number of choices is 4)

Hence ‘X’ has a binomial distribution with n= 30 and p=1/4

c) X is the same as (b), but the first 15 questions have fourchoices and the last 15 have three answer choices

p : Probability of success : Probability of getting a questionright for the first 15 questions= 1/4 (As number of choices is 4)and : Probability of getting a question right for the last 15questions= 1/3 (As number of choices is 3)

As ‘p’: probability is not same across the trails , ‘X’ does notfollow Binomial distribution.

d)

X is the number of women in a random sample of 12 students, froma class comprising 25 women and 20 men

Success : Having a women in the random sample

X :Number of successes: number of women in a randomsample

n : Number of trails : sample size of the random sample = 12

p: Probability of success : Probability of selecting a women =Number of women in the class / Total number of (women+Men) =25/25+20=25/45 = 5/9

Hence ‘X’ has a binomial distribution with n=12 and p=5/9

e)

X is the total weight of 16 randomly selected oranges

X : Number of successes ;

As X : Total weight can not be defined as a success;

‘X’ does not follow Binomial distribution.

f) X is the number of oranges, out of a random sampleof 16, that weight more than 170 grams

Success : Orange that weight more than 170 grams

X : Number of success : Number of oranges whose weight more than170 grams

n : number of trails = Number of oranges in the random sample =16

p: Probability that an orange weighs more than 170 grams.(Notknown)

Therefore ‘X’ has a binomial distribution with n= 12 and p :Probability that an orange weighs more than 170 grams: notknown