Question

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

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

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

c) X is the same as (b), but the first 15 questions have four choices 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 1

Binomial Distribution

If 'X' is the random variable representing the number of successes, the probability of getting ‘r’ successes and ‘n-r’ failures, in 'n' trails, ‘p’ probability of success ‘q’=(1-p) is given by the probability function

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 gets right on a 30-question test, when each question has four choices and the student is completely guessing on each question.

Success : questions a student gets right

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

n : Number of trails : 30-question test

p: Probability of success : Probability of getting a question right = 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 four choices and the last 15 have three answer choices

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

As 'p': probability is not same across the trails , 'X' does not follow Binomial distribution.

d)

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

Success : Having a women in the random sample

X :Number of successes:   number of women in a random sample

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 sample of 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 than 170 grams

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

p: Probability that an orange weighs more than 170 grams.(Not known)

Therefore 'X' has a binomial distribution with n= 12 and p : Probability that an orange weighs more than 170 grams: not known