Probability of a specific sum rolling a pair of dice.

Question

Rolling a pair of six-sided dice, what is the probability that you will get a sum of 4?

 

I thought the answer was 4/36 = 1/9 because each die is a mutually exclusive event 1-3, 3-1, 2-2, 2-2 but are 2-2 and 2-2 truly mutually exclusive? They're the exact same number on each die after all... so maybe the probability is 3/36 = 1/12?

Answer 1

First die       second die                Sum

1                 1,2,3,4,5, 6              2,3,4,5,6,7

2                 1,2,3,4,5, 6              3,4,5, 6,7,8

3                 1,2,3,4,5, 6              4,5,6,7,8,9

4                 1,2,3,4,5, 6              5,6,7,8,9,10

5                 1,2,3,4,5, 6              6,7,8,9,10,11

6                 1,2,3,4,5, 6              7,8,9,10,11,12

 

As you can see, there are only three sums of 4. 2+2 is counted once.

Probability = 3/36

Answer 2

Yes, 2-2 should only be counted once since they are the same pair of outcomes.

Answer 3

Probability is the ratio of (number of successful events) divided by (number of total events).

 

With two "fair" dice having numbers 1-6, there are a total of 6*6=36 total events.  How many of these 36 have a sum of 4?  Well, it might help to list them:

1 - 1  no

1 - 2  no

1 - 3  yes

1 - 4  no

. . .

2 - 2 yes

. . .

3 - 1 yes

. . .

6 - 6 no

 

There are exactly 3 times where the sum is 4 (the success criteria).  The case of 2-2 means that when the first die has a 2 and the second die also has a 2.  You would not call 1-3 and 3-1 the same outcome, would you?

 

The probability of the sum being 4 is 3/36 = 1/12.