RD Chapter 33 Probability Ex 33.2 Solutions
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.
Answer - 1 : -
Given: A coin is tossed.
When a coin is tossed, there will be two possible outcomes, Head (H) and Tail (T).
Since, the no. of elementary events is 2 {H, T}
We know, 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
∴ 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?
Answer - 2 : -
Given: Two coins are tossed once.
We know, when two coins are tossed then the no. of possible outcomes are 22 = 4
So, the Sample spaces are {HH, HT, TT, TH}
∴ 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?
Answer - 3 : -
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, according to the question,
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
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) Here, We know, if an event has only one sample point of a sample space, then it is called elementary events.
So, 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}
So, 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 .
Answer - 4 : -
Given: A dice isthrown once.
Let us find the givenevents, and also find A ∪ B,A ∩ B, B ∩ C, E ∩ F, D ∩ F and
S = {1, 2, 3, 4, 5, 6}
According to thesubparts of the question, we have certain events as:
(i) A = getting anumber below 7
So, the sample spacesfor A are:
A = {1, 2, 3, 4, 5, 6}
(ii) B = Getting anumber greater than 7
So, the sample spacesfor B are:
B = {Փ}
(iii) C = Gettingmultiple of 3
So, the Sample spaceof C is
C = {3, 6}
(iv) D = Getting anumber less than 4
So, the sample spacefor D is
D = {1, 2, 3}
(v) E = Getting aneven number greater than 4.
So, the sample spacefor E is
E = {6}
(vi) F = Getting anumber not less than 3.
So, the sample spacefor F is
F = {3, 4, 5, 6}
Now,
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.
Answer - 5 : -
Given: Three coins aretossed.
When three coins aretossed, then the sample space is
S = {HHH, HHT, HTH,HTT, THH, THT, TTH, TTT}
Now, the subparts are:
(i) The two eventswhich are mutually exclusive are when,
A: getting no tails
B: getting no heads
Then, A = {HHH} and B= {TTT}
So, the intersectionof this set will be null. Or, the sets are disjoint.
(ii) Three eventswhich are mutually exclusive and exhaustive are:
A: getting no heads
B: getting exactly onehead
C: getting at leasttwo head
So, A = {TTT} B ={TTH, THT, HTT} and C = {HHH, HHT, HTH, THH}
Since, A ⋃ B = B ⋂ C = C ⋂ A = Փ and
A⋃ B⋃ C = S
(iii) The two eventsthat are not mutually exclusive are:
A: getting three heads
B: getting at least 2heads
So, A = {HHH} B ={HHH, HHT, HTH, THH}
Hence, A ⋂ B = {HHH} = Փ
(iv) The two eventswhich are mutually exclusive but not exhaustive are:
A: getting exactly onehead
B: getting exactly onetail
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?
Answer - 6 : -
Given: A dice is thrown twice. And each time number appearing on it is recorded.
When the dice is thrown twice then the number of sample spaces are 62 = 36
Now,
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)}
Since, 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)}
∴ (AՌB) = Փ and (AՌC) ≠ Փ, A and B are mutually exclusive, but A and C are not.