# 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.4

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## Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.4

### 2nd PUC Maths Probability NCERT Text Book Questions and Answers Ex 13.4

Question 1.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
(i)

 x 0 1 2 P(X) 0.4 0.4 0.2

 X 0 1 2 T P(X) 4 4 2 1

(ii)

 X 0 1 2 3 4 P(X) 0.1 0.5 0.2 -0.1 0.3

 X 0 1 2 3 T P(X) 0.1 0.5 0.2 -0.1 1

It is not P.d as P(X) < 0 when x = 3 (iii)

 Y -1 0 1 P(Y) 0.6 0.1 0.2

 Y -1 0 1 T P(Y) 0.6 0.1 0.2 .9

It is not P.d as ΣP(X) = .9 ≠1

(iv)

 Z 3 3 1 0 -1 P(Z) 0.3 0.2 0.4 0.1 0.05

 Z 3 3 1 0 T P(Z) 0.3 0.2 0.4 0.1 1.05

It is not a P.d as ΣP(X) = 1.05 > 1

Question 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 ?
X can take the values 0, 1, 2 It is a Random variable because the experiment is a random experiment.

Question 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?
When m = 6, n = 0  ⇒ |m-n| = 6
m = 5,n = 1                  |m-n| = 4
m = 4, n = 2                 |m-nl=2
m = 3,n = 3                  |m-n| = 0
m = 2, n = 4                 |m-n|=2
m = 1,n = 5                  |m-n| = 4
m = 0, n = 6                 |m-n|=6
∴ hence X can take values 0, 2, 4, 6.

Question 4.
Find the probability distribution of
(i) number of heads in two tosses of a coin.
(ii) number of tails in the simultaneous tosses of three coins.
(iii) number of heads in four tosses of a coin.
(i) When a coin is tossed twice, the number of heads may be 0, 1,2
Sample space {HH, HT, TH, TT}  (ii) When a coin is tossed thrice, the number of heads are 0, 1, 2, 3
Sample space {HHH, HHT, HTH, HTT, ITT, TTH, THT, THH (iii) When coin is tossed four times, the number of heads be 0, 1, 2, 3,4 Sample space {HHHH, HHHT, HHTH, HHTT, HTHT, HTHH, HTTH, TTTT, TTTH, HTTT, TTHT, TTHH, THTH, THTT, THHT, THHH] Question 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
Ans:
(i) Let the random variable be X no of successes then X can take the value 0,1,2. (ii) Let X be random variable, then X can take values
O (no sixes) & 1 (at least 1 six)
P (x = 0) = P (no six)
= [1-P (getting a six)] x [1 – P (getting a six on single throw] Question 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.
Let X denote the no. of defective bulbs P denote the probability of obtaining defective bulb  Question 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.
Let X denote the number of tails in two tosses of a coin
∴ X can assain the values 0, 1,2
Let P be the probability getting head, q the tail  Question 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 2k2 7k2+k

Determination
(i) K
(ii) P(X<3)
(iii) P(X>6)
(iv) P(0<X<3) Question 9.
The random variable X has a probability distribution P(X) of the following form, where k is some number:
$$P(X)=\left\{\begin{array}{ll}{k,} & {\text { if } \quad x=0} \\{2 k,} & {\text { if } x=1} \\{3 k,} & {\text { if } x=2} \\{0,} & {\text { otherwise }}\end{array}\right.$$
(a) Determine the value of k
(b) Find P (X < 2), P (X≤2), P (X≥2)
(a) In P.d. Σ XP(X)= 1
6k = 1 ⇒ $$k=\frac{1}{6}$$  Question 10.
Find the mean number of heads in three tosses of a fair coin.
If three coins are tossed, Let X denote the number of heads.
X can take the values 0, 1, 2, 3 Question 11.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Let X denote the number of sires. Then X can take values 0,1,2 Question 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).
Ans:
X can take the values 2, 3, 4, 5, 6 Question 13.
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Ans:
Let X denote the sum of the numbers obtained
∴  X can take the values 2, 3,4, 5,6,7, 8,9, 10, 11, 12 Question 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.
Ans:
Let X denote the age of the students.
Then X can take the values 14,15,16,17, 18,19,20, 21 Question 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).  Choose the correct answer in each of the following:

Question 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
(A) 1
(B) 2
(C) 5
(D) 1 (A) $$\frac{37}{221}$$
(B) $$\frac{5}{13}$$
(C) $$\frac{1}{13}$$
(D) $$\frac{2}{13}$$ 