Class 11 RD Sharma Solutions - Chapter 33 Probability - Exercise 33.2

Question 1: A coin is tossed. Find the total number of elementary events and also the total number of events associated with the random experiment.

Solution:

Given: 

A coin is tossed.

When a coin is tossed, there will be two possible outcomes, that is Head (H) and Tail (T).

Since, the number of elementary events is 2-{H, T}

as we know that, if there are 'n' elements in a set, then the number of total element in its subset is 2n.

So, the total number of the experiment is 4,

There are 4 subset of S = {H}, {T}, {H, T} and Փ

Therefore,

There are total 4 events in a given experiment.

Question 2: List all events associated with the random experiment of tossing of two coins. How many of them are elementary events?

Solution:

Given: 

Two coins are tossed once.

As we know that, when two coins are tossed then the number of possible outcomes are 2² = 4

So, 

The Sample spaces are {HH, HT, TT, TH}

Therefore, 

There are total 4 events associated with the given experiment.

Question 3: Three coins are tossed once. Describe the following events associated with this random experiment:

A = Getting three heads, B = Getting two heads and one tail, C = Getting three tails, D = Getting a head on the first coin.

(i) Which pairs of events are mutually exclusive?

(ii) Which events are elementary events?

(iii) Which events are compound events?

Solution:

Given: 

There are three coins tossed once.

When three coins are tossed, then the sample spaces are:

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

So, as the question says,

A = {HHH}

B = {HHT, HTH, THH}

C = {TTT}

D = {HHH, HHT, HTH, HTT}

Now, 

A⋂ B = Փ,

A ⋂ C = Փ,

A ⋂ D = {HHH}

B ⋂ C = Փ,

B ⋂ D = {HHT, HTH}

C ⋂ D = Փ

As we know that, if the intersection of two sets are null or empty it means both the sets are Mutually Exclusive.

(i) Events A and B, Events A and C, Events B and C and events C and D are mutually exclusive.

(ii) Now, as we know that, if an event has only one sample point of a sample space, then it is called elementary events.

Thus, A and C are elementary events.

(iii) If there is an event that has more than one sample point of a sample space, it is called a compound event.

Since, B ⋂ D = {HHT, HTH}

Thus, B and D are compound events.

Question 4: In a single throw of a die describe the following events:

(i) A = Getting a number less than 7

(ii) B = Getting a number greater than 7

(iii) C = Getting a multiple of 3

(iv) D = Getting a number less than 4

(v) E = Getting an even number greater than 4.

(vi) F = Getting a number not less than 3.

Also, find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and  

Solution:

Given: 

A dice is thrown once.

Now, find the given events, and also find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and \overline F

S = {1, 2, 3, 4, 5, 6}

According to the question, we have certain events as:

(i) A = Getting a number below 7

Thus, the sample spaces for A are:

A = {1, 2, 3, 4, 5, 6}

(ii) B = Getting a number greater than 7

Thus, the sample spaces for B are:

B = {Փ}

(iii) C = Getting multiple of 3

Thus, the Sample space of C is

C = {3, 6}

(iv) D = Getting a number less than 4

Thus, the sample space for D is

D = {1, 2, 3}

(v) E = Getting an even number greater than 4.

Thus, the sample space for E is

E = {6}

(vi) F = Getting a number not less than 3.

Thus, the sample space for F is

F = {3, 4, 5, 6}

Here,

A = {1, 2, 3, 4, 5, 6} and B = {Փ}

A ⋃ B = {1, 2, 3, 4, 5, 6}

A = {1, 2, 3, 4, 5, 6} and B = {Փ}

A ⋂ B = {Փ}

B = {Փ} and C = {3, 6}

B ⋂ C = {Փ}

F = {3, 4, 5, 6} and E = {6}

E ⋂ F = {6}

E = {6} and D = {1, 2, 3}

D ⋂ F = {3}

and, for \overline F = S - F\\S= \{1,2,3,4,5,6\} and F =\{3,4,5,6\}\\\overline F = \{1,2\}\\Therefore,\\These\ are\ the\ events\ for\ given\ experiment.

Question 5: Three coins are tossed. Describe

(i) two events A and B which are mutually exclusive.

(ii) three events A, B, and C which are mutually exclusive and exhaustive.

(iii) two events A and B which are not mutually exclusive.

(iv) two events A and B which are mutually exclusive but not exhaustive.

Solution:

Given: 

Three coins are tossed.

When three coins are tossed, then the sample spaces are

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Here, 

(i) The two events which are mutually exclusive are when,

A: getting no tails

B: getting no heads

Then, 

A = {HHH} and B = {TTT}

So, the intersection of this set will be null. Or, the sets are disjoint.

(ii) Three events which are mutually exclusive and exhaustive are:

A: getting no heads

B: getting exactly one head

C: getting at least two head

Thus, 

A = {TTT} 

B = {TTH, THT, HTT} and, 

C = {HHH, HHT, HTH, THH}

Hence, 

A ⋃ B = B ⋂ C = C ⋂ A = Փ and

A⋃ B⋃ C = S  

(iii) The two events that are not mutually exclusive are:

A: getting three heads

B: getting at least 2 heads

So, 

A = {HHH} 

B = {HHH, HHT, HTH, THH}

Hence, A ⋂ B = {HHH} = Փ

(iv) The two events which are mutually exclusive but not exhaustive are:

A: getting exactly one head

B: getting exactly one tail

So, 

A = {HTT, THT, TTH} and B = {HHT, HTH, THH}

It is because A ⋂ B = Փ but A⋃ B ≠ S  

Question 6: A die is thrown twice. Each time the number appearing on it is recorded. Describe the following events:

(i) A = Both numbers are odd.

(ii) B = Both numbers are even

(iii) C = sum of the numbers is less than 6.

Also, find A ∪ B, A ∩ B, A ∪ C, A ∩ C. Which pairs of events are mutually exclusive?

Solution:

Given: 

A dice is thrown twice. Each time number appearing on it is recorded.

When a dice is thrown twice then the number of sample spaces are 6² = 36

Here,

The possibility both odd numbers are:

A = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

Thus, possibility of both even numbers is:

B = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}

And, possible outcome of sum of the numbers is less than 6.

C = {(1, 1)(1, 2)(1, 3)(1, 4)(2, 1)(2, 2)(2, 3)(3, 1)(3, 2)(4, 1)}

Hence,

(AՍB) = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5) (2, 2)(2, 4)(2, 6)(4, 2)(4, 4)(4, 6)(6, 2)(6, 4)(6, 6)}

(AՌB) = {Փ}

(AUC) = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5) (1, 2)(1, 4)(2, 1)(2, 2)(2, 3)(3, 1)(3, 2)(4, 1)}

(AՌC) = {(1, 1), (1, 3), (3, 1)}

Therefore,

(AՌB) = Փ and (AՌC) ≠ Փ, A and B are mutually exclusive, but A and C are not.

Question 7: Two dice are thrown. the events A, B, C, D, E, and F are described as follows:

A=Getting an even number on the first die.

B=Getting an odd number on the first die.

C=Getting at most 5 as sum of the number on the two dice.

D=Getting the sum of the numbers on the dice greater than 5 but less than 10.

E=Getting at least 10 as the sum of the numbers on the dice.

F=Getting an odd number on one of the dice.

(i) Describe the following events:

A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false:

(a) A and B are mutually exclusive

(b) A and B are mutually exclusive and exhaustive events.

(c) A and C are mutually exclusive events.

(d) C and D are mutually exclusive and exhaustive events.

(e) C, D, and E are mutually exclusive and exhaustive events.

(f) A' and B' are mutually exclusive events.

(g) A, B, F are mutually exclusive and exhaustive events.

Solution:

A = Getting an even number on the first die.

A = {(2, 1), (2, 2) (2, 3), (2, 4), (2, 5), (2, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}    

B = Getting an odd number on the first die.

B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}

C = Getting at most 5 as sum of the numbers on the two dice.

C = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}

D = Getting the sum of the numbers on the dice > 5 but < 10.

D = {(1, 5) (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 5), (3, 6), (4, 2), (4, 3), (4, 4), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3)}

F = Getting an odd number on one of the dice.

F = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)}

Its clear that A and B are mutually exclusive events and A ∩ B = ∅

B ∪ C = {(1, 1), (1,2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3,3), (3, 4), (3, 6), (5, 1), (5, 2), (5, 3), (5, 5), (5, 6), (2,1), (2,2), (2, 3), (4, 1)}

B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}

A ∩ E = {(4, 6), (6, 4), (6, 5), (6, 6)}

A ∪ F = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (5, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}

A ∩ F = {(2, 1), (2,3), (2,5), (4,1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)}

(ii)

a) True, A ∩ B = ∅

b) True, A ∩ B = ∅ and A ∪ B = S

c) False, A ∩ C ≠ ∅

d) False, A ∩ B = ∅ and A ∪ B ≠ S

e) True, C ∩ D = D ∩ E = C ∩ E = Φ and C ∪ D ∪ E = S

f) True, A' ∩ B' = ∅

g) False, A ∩ F ≠ ∅

Question 8: The number 1, 2, 3, and 4 are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly.  A person draws two slips from the box, one after the other, without replacement. describe the following events:

A=The numbers on the first slip is larger than the one on the second slip.

B =The number on the second slip is greater than 2

C=The sum of the numbers on the two slips is 6 or 7

D=The number on the second slips is twice that on the first slip.

Which pair(s) of events is (are) mutually exclusive?

Solution:

We have four slips of paper with numbers 1, 2, 3 & 4.

A person draws two slips without replacement.

∴ Number of elementary events = ⁴C₂

n(s)=\frac{4\times3}{2\times1}=6

A = The number on the first slip is larger than the one on the second slip

A = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}

B = The number on the second slip is greater than 2

Therefore, 

B = {(1,3), (2,3) , (1,4), (2, 4), (3, 4), (4,3)}

C = The sum of the numbers on the two slips is 6 or 7

Therefore, 

C = {(2, 4), (3, 4), (4, 2), (4, 3)}

and,

D = The number on the second slips is twice that on the first slip

D = {(1, 2), (2, 4)}

and, A and D form a pair of mutually exclusive events as A ∩ B = ∅

Question 9: A card is picked up from a deck of 52 playing cards.

(i) What is the sample space of the experiment?

(ii) What is the event that the chosen card is ace faced card?

Solution:

(i) Sample space for picking up a card from a set of 52 cards is set of 52 cards itself.

(ii) For an event of chosen card be black faced card, event is a set of jack, king, queen of spades and clubs,

Summary

Exercise 33.2 covers probability concepts, including experimental probability, theoretical probability, and probability of events. Students learn to calculate probabilities using formulas and theorems. Understanding probability is crucial for statistics, data analysis, and real-world applications.