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 

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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 

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}

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

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,