OP Malhotra Class 11 Maths Solutions Chapter 22 Probability Ex 22(a)

Interactive S Chand ISC Maths Class 11 Solutions Chapter 22 Probability Ex 22(a) engage students in active learning and exploration.

S Chand Class 11 ICSE Maths Solutions Chapter 22 Probability Ex 22(a)

Question 1.
What do you mean by Random Experiment ? Give two illustrations. Define sample space associated with a random experiment. Give an example.
Solution:
Random Experiment: An experiment whose all outcomes are known in advance but outcomes of experiment cannot be predictable.
e.g. : Tossing a coin, its outcomes are known i.e. either head or tail but we can’t predict the outcome i.e. on tossing a coin, we can’t predict whether head comes or tail. e.g. : throwing a dice, outcomes are known i.e. {1, 2, 3, 4, 5, 6} but we can’t predict the outcomes which number comes.
Sample space : The set of all outcomes of a random experiment is called sample space.
e.g. : On tossing a coin, outcomes are either head or tail.
∴ Sample space = {H, T}

Question 2.
What is the resulting sample space if
(i) one coin is tossed ;
(ii) two coins are tossed simultaneously ;
(iii) three coins are tossed simultaneously ?
Solution:
(i) When one coin is tossed
Then sample space S = {H, T}
(ii) Two coins are tossed simultaneously
Then S = {HH, HT, TH, TT}

Question 3.
Describe the sample space of this experiment:
(i) One die is rolled ;
(ii) Two dice are rolled.
Solution:
(i) When one die is rolled
Then sample space S = {1, 2, 3,4, 5, 6}
(ii) When two dice are rolled
Then S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

Question 4.
Describe the sample space :
(i) A coin is tossed twice. If the second throw results in a tail, a die is thrown.
(ii) A coin is tossed twice. If the second throw results in a head, a die is thrown, otherwise a coin is tossed.
(iii) A coin is tossed. If it results in a head, a die is thrown. If the die shows up an even number, the die is thrown again.
Solution:
(i) S = {HH, HT1, HT2, HT3, HT4, HT5, HT6, TT1, TT2, TT3, TT4, TT5, TT6}
(ii) S = {HTH, TTH, TTT, HTT, HH1, HH2, HH3, HH4, HH5, HH6, TH1, TH2, TH3, TH4, TH5, TH6}
(iii) S = {T, HI, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}

Question 5.
A five-sided spinner is spun and a coin is tossed.
(i) Show the combined outcomes in a space diagram and in a tree diagram.
(ii) List the combined outcomes and state the number of equally likely combined outcomes.

Solution:

(ii) {(H , 1), (H, 2), (H, 3), (H, 4), (H, 5), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5)}
∴ required total no. of outcomes = 10

Question 6.
In a bag there are three balls ; one red, one blue and one yellow. A ball is selected, the colour is recorded and the ball is replaced. A second ball is then selected and the colour is recorded.
(i) Show in a space diagram and in a tree diagram all the possible combined outcomes.
(ii) List these combined outcomes and state the number of equally likely combined outcomes.
Solution:

(ii) {(R. R), (R, B), (R, Y), (B, R), (B, B), (B, Y), (Y, R), (Y, B), (Y, Y)}

Question 7.
Satish and Mukesh who live in London wish to go on a holiday to France. They can travel to the coast by car, coach or train, and then cross the channel by ferry, train, helicopter or hovercraft.
(i) In a space diagram and in a tree diagram show all the combined outcomes of the different ways they could travel to France.
(ii) How many different ways could they travel ?
Solution:

(ii) Thus required no. of ways = 4 + 4 + 4 = 12

Question 8.
From a group of 2 men and 3 women, two persons are selected. Describe the sample space of the experiment. If E is the event in which one man and one woman are selected, then which are the cases favourable to E ?
Solution:
Given we have group of 2 men {M₁, M₂} and 3 women {W₁, W₂, W₃}
∴ Sample space = {M₁M₂, M₁W₁, M₁W₂, M₁W₃, M₂W₁, M₂W₂, M₂W₃, W₁W₂, W₂W₃, W₁W₃} Given E : event in which one man and one woman be selected
∴ favourable cases to E = {M₁W₁, M₁W₂, M₁W₃, M₂W₁, M₂W₂, M₂W₃}

Question 9.
A coin is tossed. If it results in a head, a coin is tossed, otherwise a die is thrown. Describe the following events:
(i) A: getting at least one head ;
(ii) B : getting an even number ;
(iii) C : Getting a tail;
(iv) D : getting a tail and an odd number.
Solution:
Sample space = {HH, HT, T1, T2, T3, T4, T5, T6}
(i) A = {HH, HT}
(ii) B = {T2, T4, T6}
(iii) C = {HT, Tl, T2, T3, T4, T5, T6}
(iv) D = {T1, T3, T5}

Question 10.
A coin and a die are tossed. Describe the following events.
(i) A : getting a head and an even number ;
(ii) B : getting a prime number ;
(iii) C : getting a tail and an odd number;
(iv) D : getting a head or a tail.
Solution:
When a coin and a dice are thrown
Then sample space S = {H1, H2, H3, H4, H5, H6, Tl, T2, T3, T4, T5, T6}
(i) A = {H2, H4, H6}
(ii) B = {H2, H3, H5, T2, T3, T5}
(iii) C = {T1, T3, T5}
(iv) D = {H1, H2, H3, H4, H5, H6, Tl, T2, T3, T4, T5, T6}

Question 11.
A fair coin is tossed. If it shows a head, we draw a ball from a bag consisting of 3 distinct red and 4 distinct black balls, if it shows a tail, we throw a fair die. Draw a tree diagram to show all the possible outcomes and obtain .he sample space. What are sets representing the following events:
(i) the ball drawn is black ;
(ii) the coin shows tail.
Solution:

∴ S = {HR₁, HR₂, HR₃, HB₁, HB₂, HB₃, HB₄, T₁, T₂, T₃, T₄, T₅, T₆)
(i) {(H, B₁), (H, B₂), (H, B₃), (H, B₄)}
(ii) {(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}

Question 12.
Two dice are rolled. A is the event that the sum of the numbers shown on the two dice is 5. B is the event that at least one of the dice shows up a 3. Are the two events A and B (i) mutually exclusive, (ii) exhaustive ? Give arguments in support of your answer.
Solution:
When two dice are rolled
Then S = {(1, 1), (1,2), (1,3), (1,4), (1, 5), (1,6), (2,1), (2,2), (2,3), (2,4), (2, 5), (2,6), (3,1), (3,2), (3,3), (3,4), (3, 5), (3,6), (4, 1), (4,2), (4, 3), (4,4), (4, 5), (4, 6), (5, 1), (5,2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ A = {(1,4), (2, 3), (3, 2), (4,1)}
B = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)} .
Here A ∩ B = {(2, 3), (3, 2)} ≠ Φ
Thus A and B are not mutually exclusive events.
Here, A∪B ≠ S .
∴ A and B are not exhaustive events.