**Difficulty: Tricky**

Alex is playing a dice game with 2 standard 6-sided dice, one red and one blue. Each die has faces numbered from 1 to 6. Alex rolls both dice and adds up the numbers on the top faces. After a while, Alex notices that the sums are always between 2 and 12 and this means there are 6 possible even sums (2, 4, 6, 8, 10, and 12) and only 5 possible odd sums (3, 5, 7, 9, and 11).

What is the probability of rolling an odd sum?

**Scroll down or click for a hint, or the answer!
**

How many possible rolls are there? Remember that each of the 2 dice have 6 possible faces so we multiply 6 x 6 to get 36 total possible rolls. Written out as (red die, blue die), that looks like:

(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).

Now, how many of those rolls sum to odd numbers? The first odd number is 3, and only 2 rolls, (1,2) and (2,1), sum to 3.

What about the other odd numbers?

The chance of rolling an odd sum is ½ or 50%.

You can go about showing this in 2 ways. First, you can calculate the number of rolls that sum to odd numbers:

3: can be rolled as (1,2) or (2,1). So 2 rolls make this odd sum.

5: (1,4), (2,3), (3,2), or (4,1) = 4 rolls

7: (1,6), (2,5), (3,4), (4,3), (5,2) or (6,1) = 6 rolls

9: (3,6), (4,5), (5,4), or (6,3) = 4 rolls

11: (5,6) or (6,5) = 2 rolls

Add them together to find the total odd rolls: 2 + 4 + 6 + 4 + 2 = 18. Remember that each die has 6 possible faces so there are 6 x 6 = 36 possible rolls. In other words, there are 36 different possible rolls and 18 of them sum to an odd number. As a fraction, that’s 18/36 = ½ = 50% odd rolls.

Don’t believe us? We can check the even numbers, too:

2: (1,1) = 1 roll

4: (1,3), (2,2), (3,1) = 3 rolls

6: (1,5), (2,4), (3,3), (4,2), (5,1) = 5 rolls

8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 rolls

10: (4,6), (5,5), (6,4) = 3 rolls

12: (6,6) = 1 roll

The total number of even rolls = 1 + 3 + 5 + 5 + 3 + 1 = 18. Again, 18 even rolls out of 36 possible rolls makes the chance of rolling an even sum equal to 18/36 = ½ or 50%.

Alternatively, we can visualise this problem using a 6 x 6 table where the value of the red die is across and the value of the blue die is down. The rest of the table colour-codes their sums so that green is even and yellow is odd:

There are 3 even and 3 odd numbers in each row and there are 6 rows total, meaning there are 3 x 6 = 18 evens and 3 x 6 = 18 odds total in the table. This confirms what we see visually: the table looks half and half.

If you look closer, you can see that our first method counted evens and odds along the diagonals. Both get us to the same 50/50 answer.

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30 August, 2023 at 1:02 pm

The first roll is either even or odd, with equal chance. So is the second roll. To give an even total, both rolls must be even or both rolls must be odd. Probability of both even is half x half = one-quarter. Probability of both odd is the same. So, the probability of an even total is a quarter plus a quarter, that is one-half. The alternative is an odd total, and that must account for all the remaining probability and is also one-half. The problem does not need a sample space and examination of every way the dice can fall!

30 August, 2023 at 1:15 pm

An excellent and concise solution! Thanks for pointing this one out, Paul.