Mathematics Part-II

Book: Mathematics Part-II

Chapter: 13. Probability

Subject: Maths - Class 12th

Q. No. 4 of Exercise 13.4

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4

Find the probability distribution of

number of tails in the simultaneous tosses of three coins.

three coins are tossed simultaneously. Hence, the sample space of the experiment is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}


X represents the number of tails.


As we see, X is a function on sample space whose range is {0, 1, 2, 3}.


Thus, X is a random variable which can take the values 0, 1, 2 or 3.


P(X = 0) = P(HHH)


P(X = 1) = P(HHT) + P(HTH) + P(THH)


P(X = 2) = P(HTT) + P(THT) + P(TTH)


P(X = 3) = P(TTT)


Hence, the required probability distribution is,


X



0



1



2



3



P(X)



1/8



3/8



3/8



1/8



4

Chapter Exercises

Exercise 13.1
Exercise 13.2
Exercise 13.3
Exercise 13.4
Exercise 13.5
Miscellaneous Exercise

More Exercise Questions

1

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
1

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
1

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
1

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
2

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?
3

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?
4

Find the probability distribution of

number of heads in two tosses of a coin.
4

Find the probability distribution of

number of tails in the simultaneous tosses of three coins.
4

Find the probability distribution of

number of heads in four tosses of a coin.
5

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
6

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.
7

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.
8

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)
9

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).
10

Find the mean number of heads in three tosses of a fair coin.
11

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
12

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).
13

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
14

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.
15

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).
16

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
17

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