Question

Question 7 Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice ≤ 5

State true or false: (give reason for your answer)

(i) A and B are mutually exclusive

(ii) A and B are mutually exclusive and exhaustive

(iii) A = B '

(iv) A and C are mutually exclusive

(v) A and B ' are mutually exclusive

(vi) A ', B ', C are mutually exclusive and exhaustive.

Class X1 - Maths -Probability Page 393

Answer 1

If two dice are thrown then,total number of possible outcomes S = 6 × 6 = 36
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 = getting an even number on the first die
= {(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 on the first die.
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3),(3,4), (3,5), (5,1),(5,2),(5,3), (5,4),(5,5),(5,6)}

C = getting the sum of the numbers on the die ≤ 5
= {(1,1), (1,2), (1,3), (1,4), (2,1),(2,2),(2,3),(3,1),(3,2),(4,1)}

(i) A' : A' is the events not getting an even number on the first dice .
or we can say that A' is the events getting an odd number on the first dice . e.g., A' = B

(ii) not B { B'} : B' is the event not getting an odd number on the first die . or we can say that B' is the event getting an even number on the first die.e.g., B' =A

(iii) A or B ( A U B ) : A or B (A U B) is the event getting all elements of A and B .e.g., A or B is the events getting an odd number on the first die and an even number on the first die.

(iv) A and B ( A ∩ B) : A and B is the event getting common elements of A and B .

(v) A but not C ( A - C) : Elements which are in A but in C
= { (2,4), (2,5), (2,6), (4,2) , (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5),(6,6)}

(vi) B or C ( B ∪ C ) : elements which are both in B and C .
=(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),(2,2),(2,3),(4,1)}

(vii) B and C ( B∩C) : elements which are common in both B and C .
= {(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}

(viii) A ∩ B' ∩ C' : A or not B or not C = A ∩ (B U C)'
= {(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1)(6,2),(6,3)(6,4),(6,4),(6,5),(6,6)}




(i) True,
∵ A = getting an even number on the first die.
B = getting an odd number on the first die.
A ∩ B = Ф
∴ A and B are mutually exclusive events

(ii) True,
∵A U B = S { if A union B is equal to whole sample space then, it is exhaustive .}
∵ A = getting an even number on the first die.
B = getting an odd number on the first die.
∴ A ∩ B = Ф, ∴ A and B are mutually exclusive.

(iii)True,
A = B' { see answer of above question (ii) }

(iv) False ,
∵ A ∩ C = { (2,1),(2,2),(2,3),(4,1)} ≠ Ф
∴ A and C are not mutually exclusive.

(v) False ,
∵ B' = A
and A ∩ A = A ≠Ф
Hence, A and B' are not mutually exclusive.

(vi) False,
∵ A' = B , B' = A
∴ A' ∩ B' ∩ C = B ∩ A ∩ C = Ф
but A'∩ C = B ∩ C ≠ Ф so, A' , B' and C are not mutually exclusive..

A' ∪ B' ∪ C = B ∪ A ∪ C = S { sample space }
Hence, A' , B' and C are exhaustive .