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

Book: Mathematics Part-II

Chapter: 13. Probability

Subject: Maths - Class 12^th

Q. No. 9 of Exercise 13.4

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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).

Given: A random variable X with its probability distribution.

(a) As we know the sum of all the probabilities in a probability distribution of a random variable must be one.

Hence the sum of probabilities of given table:

⇒ k + 2k + 3k + 0 = 1

⇒ 6k = 1

(b) (i) P(X < 2) = ?

P(X < 2) = P(X = 0) + P(X = 1)

= k + 2k

= 3k

(ii) P(X ≤ 2) = ?

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

= k + 2k + 3k

= 6k

(iii) P(X ≥ 2) = ?

P(X ≥ 2) = P(X = 2) + P(X > 2)

= 3k + 0

= 3k

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. 9 of Exercise 13.4

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).

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. 9 of Exercise 13.4

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).

Chapter Exercises

More Exercise Questions

Subject: Maths - Class 12^th

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).