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The Galileo paradox

With two dice rolled 100 times, which total appears more often, 9 or 10? Most people guess they're equal, or lean toward 10. Think about it before moving forward.

What do you think?
Which sum is more common?

Roll the dice and watch the bar chart build up. 9 appears more often than 10. Hit "×100" a few times to see the pattern clearly.

Roll 2 dice
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+
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=?

Why 9 beats 10

If you count systematically, there are 4 ways to make 9 — (3,6), (4,5), (5,4), (6,3) — but only 3 ways to make 10: (4,6), (5,5), (6,4). That single extra combination is the whole story.

What do you think?
How many ways can two dice sum to 9?
Enter a whole number

Notice that (5,5) only counts once because you can't flip it. But (4,6) and (6,4) are different: Die 1 showing 4 and Die 2 showing 6 is distinct from Die 1 showing 6 and Die 2 showing 4.

P(rolling 9) = 4/36 = 1/9 ≈ 11.1%

P(rolling 10) = 3/36 = 1/12 ≈ 8.3%

This 6×6 grid shows every possible outcome. Click a sum to see which cells match. 7 has the most, 2 has the fewest.

Dice Pair Grid
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
Ways to make sum
4
Probability
4/36
P(sum = 9) = 4/36 ≈ 11.1%

Can you find all the pairs for a given sum? Click every cell that produces the target.

Find the Pairs
Click all cells that sum to 8
1
2
3
4
5
6
1
2
3
4
5
6
How many ways can two dice sum to 7? (whole number)
How many ways can two dice sum to 2? (whole number)

Galileo and the Grand Duke

Around 1620, the Grand Duke of Tuscany noticed players betting on 9 were winning more than those betting on 10. He asked Galileo to investigate.

Galileo counted every outcome and found the mistake: gamblers were treating (3,6) and (6,3) as the same outcome.

What do you think?
What mistake were gamblers making?

Counting is harder than it looks.

Total possible outcomes for two dice? (6 faces each) (whole number)
P(sum = 7)? (fraction, e.g. 2/7)