From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Book: Mathematics Part-II

Chapter: 13. Probability

Subject: Maths - Class 12^th

Q. No. 6 of Exercise 13.4

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From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Given: A lot of 30 bulbs which include 6 defectives.

Then number of non-defective bulbs = 30 – 6 = 24

As 4 bulbs are drawn at random with replacement.

Let X denotes the number of defective bulbs from the selected bulbs.

Clearly, X can take the value of 0, 1, 2, 3 or 4.

P(X = 0) = P(4 are non defective and 0 defective)

P(X = 1) = P(3 are non defective and 1 defective)

P(X = 2) = P(2 are non defective and 2 defective)

P(X = 3) = P(1 are non defective and 3 defective)

P(X = 4) = P(0 are non defective and 4 defective)

Hence, the required probability distribution is,

X	0	1	2	3	4
P(X)

Chapter Exercises

Exercise 13.1

Exercise 13.2

Exercise 13.3

Exercise 13.4

Exercise 13.5

Miscellaneous Exercise

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

X

0

1

2

P(X)

0.4

0.4

0.2

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

X

0

1

2

3

4

P(X)

0.4

0.5

0.2

-0.1

0.3

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Y

-1

0

1

P(Y)

0.6

0.1

0.2

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Z

3

2

1

0

-1

P(Z)

0.3

0.2

0.4

0.1

0.05

An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable?

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?

Find the probability distribution of
number of heads in two tosses of a coin.

Find the probability distribution of
number of tails in the simultaneous tosses of three coins.

Find the probability distribution of
number of heads in four tosses of a coin.

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4

(ii) six appears on at least one die

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

A random variable X has the following probability distribution:

X

0

1

2

3

4

5

6

7

P(X)

0

k

2k

2k

3k

K²

2 K²

7 K² + k

Determine

(i) k (ii) P(X < 3)

(iii) P(X > 6) (iv) P(0 < X < 3)

The random variable X has a probability distribution P(X) of the following form, where k is some number:

(a) Determine the value of k.

(b) Find P (X < 2), P (X ≤ 2), P(X ≥ 2).

Find the mean number of heads in three tosses of a fair coin.

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X).

Book: Mathematics Part-II

Chapter: 13. Probability

Subject: Maths - Class 12^th

Q. No. 6 of Exercise 13.4

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Chapter Exercises

Exercise 13.1

Exercise 13.2

Exercise 13.3

Exercise 13.4

Exercise 13.5

Miscellaneous Exercise

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

X

0

1

2

P(X)

0.4

0.4

0.2

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

X

0

1

2

3

4

P(X)

0.4

0.5

0.2

-0.1

0.3

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Y

-1

0

1

P(Y)

0.6

0.1

0.2

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Z

3

2

1

0

-1

P(Z)

0.3

0.2

0.4

0.1

0.05

An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable?

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?

Find the probability distribution of
number of heads in two tosses of a coin.

Find the probability distribution of
number of tails in the simultaneous tosses of three coins.

Find the probability distribution of
number of heads in four tosses of a coin.

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4

(ii) six appears on at least one die

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

A random variable X has the following probability distribution:

X

0

1

2

3

4

5

6

7

P(X)

0

k

2k

2k

3k

K²

2 K²

7 K² + k

Determine

(i) k (ii) P(X < 3)

(iii) P(X > 6) (iv) P(0 < X < 3)

The random variable X has a probability distribution P(X) of the following form, where k is some number:

(a) Determine the value of k.

(b) Find P (X < 2), P (X ≤ 2), P(X ≥ 2).

Find the mean number of heads in three tosses of a fair coin.

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X).

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is

Book: Mathematics Part-II

Chapter: 13. Probability

Subject: Maths - Class 12th

Q. No. 6 of Exercise 13.4

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Chapter Exercises

More Exercise Questions

Subject: Maths - Class 12^th