Here are the possible outcomes when a pair of fair dice is rolled.
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
in a roll of a pair of fair dice, what is the probability of the outcome being either a multiple of 3 or an even number? are these events mutually exclusive?
, mutually exclusive
, not mutually exclusive
, mutually exclusive
, not mutually exclusive

I don't understand the big table of numbers at the top at all,
and don't see how it relates to the question.

There are 36 possible outcomes for the roll of a pair of dice.
According to your specifications, the successes are:

-- Multiples of 3:
       1 ... 2    (3)
       2 ... 1      
       3 ... 3    (6)
       1 ... 5
       5 ... 1
       2 ... 4
       4 ... 2
       3 ... 6    (9)
       6 ... 3
       4 ... 5
       5 ... 4
       6 ... 6    (12)
(12 different outcomes)

-- Even numbers:
       1 ... 1   (2) 
       1 ... 3    (4)
       3 ... 1
       2 ... 2
       3 ... 3    (6)
       1 ... 5
       5 ... 1
       2 ... 4
       4 ... 2  
       2 ... 6    (8)
       6 ... 2
       3 ... 5
       5 ... 3
       4 ... 4
       4 ... 6    (10)
       6 ... 4
       5 ... 5
       6 ... 6    (12)   
(18 different outcomes)

The events are NOT mutually exclusive.
A roll of 6 (5 ways)  or 12 (1 way)  meets both requirements.

Successful outcomes:
  Multiples of 3 . . . . 12
  Even numbers . . . 18
       Duplicates . . . . 6
              
So there are 24 different successful outcomes.

Probability  = (24) / (36)  =  2/3  =  (66 and 2/3) %