Suppose a die is tossed. Write two events which are:
a. Exhaustive and mutually exclusive.
b. Exhaustive but not mutually exclusive.
c. Exhaustive but not equally likely
My input :
a. Events are :
$A=$ Getting even number when die is thrown.
$B=$ Getting odd number when die is thrown.
b. Events are :
$A=$ Getting number less than $5$ when die is thrown.
$B=$ Getting multiple of $2$ when die is thrown.
Did I write them correctly?
c. I am stuck at this one. Honestly, I didn't understand the "not equally likely or equally likely". Please, someone, tell me the meaning of it and one example of it too. Don't give an example related to this particular question, after getting understanding of it I 'll try to make one by myself.
Best Answer
Let S be the universal set
Set S :{1,2,3,4,5,6}
Set A :{your choice} $A \in\ S$
Set B :{your choice} $B \in\ S$
Exhaustive Events: You can choose any way you want to define A and B such that $A\cup\ B$ = S Example :
|A ={1} B={2,3,4,5,6}|
|A ={1,3} B={2,4,5,6}|
|A ={1,2,5,6} B={3,4}|
|A ={1,2,3,6} B={2,3,4,5}|
|A ={1,2,3} B={2,3,4,5,6}|
|A ={1,2,3,4,5} B={1,2,3,4,5,6}| .....
Mutually Exclusive : You can choose any way you want to define A and B such that $A \cap\ B$ = $\phi\ $
i.e. There should be no common element between A and B
Example :
|A ={1} B={2,3,4,5,6}|
|A ={1,3} B={2,4,5,6}|
|A ={1,2,5,6} B={3,4}|
|A ={1,2,3} B={4,5}|
|A ={1} B={2,3}|
|A ={1,6} B={2,4,5}| .....
Equally likely: It means the probability of occurring of events A and B is same
Here probability is determined by the size of set A and B.
For equally likely both sets have the same size
Examples for equally likely
|A ={1,2,3} B={2,3,4}|
|A ={1,3} B={4,5}|
|A ={1,2} B={1,2}|
|A ={1,3,5} B={2,4,6}|
|A ={1} B={6}|
|A ={1,4,5,6} B={2,3,4,5}| .....