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The dice sum race

Probability · level 2, lesson 4 of 6

A question to start

Here is a race for a group of friends. There are 11 horses, numbered 2 to 12. Each player picks a horse. Then you take turns rolling two dice and adding them. Whatever the total is, that horse moves one square forward. The first horse to reach the finish wins.

23456789101112startfinish

Which horse would you pick? Does it matter? Everyone who plays this game a few times notices the same thing: horse 7 keeps winning, and horses 2 and 12 hardly ever leave the start. Why?

The idea

Two dice: name them

The secret is in the two dice. To see it, do what you did in Fair games: give the dice names. Say one die is orange and the other is blue. Then "orange 3, blue 4" and "orange 4, blue 3" are two different ways to roll, even though both make 7.

How many ways can the two dice land? The orange die has 6 faces. For each of those, the blue die has 6 faces. So there are 6 rows of 6, which is 36 ways altogether.

Each of the 36 ways is equally likely. That is because each die is fair on its own, and the two dice do not affect each other: the blue die has no idea what the orange die did.

The table of all 36 ways

Here are all 36 ways, with the total in each square. The orange die is down the side and the blue die is along the top.

234567345678456789567891067891011789101112 123456123456 7: 6 ways6 or 8: 5 ways each2 or 12: 1 way each

Look at the orange squares. Total 7 runs right across the table from corner to corner, along a diagonal: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. That is 6 ways: one in every row.

Total 2 has only 1 way: 1 and 1, the top-left corner. Total 12 also has 1 way: 6 and 6, the bottom-right corner.

Why 7 has the most ways

There is a neat reason why 7 is the winner. Whatever the orange die shows, there is exactly one blue face that makes 7:

  • orange 1 needs blue 6;
  • orange 2 needs blue 5;
  • ... and so on, up to orange 6, which needs blue 1.

So every row of the table has a 7 in it. No other total can do that. For a total of 8, orange 1 would need blue 7, which does not exist. For a total of 2, orange 2 would already be too big. Only 7 has a partner for every face, because 7 is 1 more than 6, so each face from 1 to 6 has a partner from 6 down to 1.

The pyramid of ways

Count the ways for every total and stack them up, one square per way.

23456789101112

It makes a pyramid. The ways go up by 1 from total 2 to total 7, and down by 1 from 7 to 12: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. Add them all: 36. ✓ (Every square of the table is counted once.)

The pyramid is the shape of the race. Horse 7 gets pushed forward on 6 of the 36 ways, horse 2 on only 1. So on average, horse 7 moves 6 times as often as horse 2 or horse 12.

What really happens in a race

Say each horse needs to move 5 squares to finish, as on the board above. A computer can play this race a million times very quickly. Here is about how often each horse wins, out of every 1000 races.

Horse 2 3 4 5 6 7 8 9 10 11 12
Wins in 1000 races (about) less than 1 9 39 99 193 320 193 99 39 9 less than 1

Three things to notice.

  1. Horse 7 wins most often: about 320 races in 1000, roughly 1 race in 3.
  2. Horse 7 does not win most races. It loses about 2 races in 3! "Most likely to win" is not the same as "likely to win". Horses 6 and 8 together win a little more often than horse 7 does.
  3. The race makes small differences bigger. On a single roll, 7 (6 ways) is only a little more likely than 6 (5 ways). But in a race, horse 7 wins about 320 times to horse 6's 193. Being a little faster at every step adds up over a whole race. And horse 2, which moves 6 times less often than horse 7, almost never wins: about once in 2000 races.

Is the race fair?

No. The horses are not equal: some have more ways than others, so whoever picks 7 has the best chance. To make a fair game with two dice, give each player a group of totals with the same number of ways. For example, one player wins on a 7 (6 ways), and the other wins on a 2, 3 or 4 (1 + 2 + 3 = 6 ways); on any other total, roll again. Both have 6 ways, so it is fair. There are more fair splits in the examples and questions below.

Worked examples

Example 1. Ways to make 5.

How many ways can two dice make a total of 5?

  1. Go down the orange die: orange 1 needs blue 4; orange 2 needs blue 3; orange 3 needs blue 2; orange 4 needs blue 1.
  2. Orange 5 or 6 is already too big (the blue die would need 0 or less).
  3. That is 4 ways. It matches the pyramid.

Example 2. Why "3 and 4" counts twice.

Tom says, "A total of 7 is 1 and 6, 2 and 5, or 3 and 4. That is only 3 ways, and 2 is 1 and 1, just 1 way. So 7 is only 3 times as likely as 2."

  1. Name the dice. "3 and 4" can be orange 3 with blue 4, or orange 4 with blue 3. Those are two different ways.
  2. The same goes for "1 and 6" and "2 and 5". So 7 has 3 × 2 = 6 ways.
  3. But "1 and 1" is just orange 1 with blue 1. Swapping does not give a new way. So 2 has 1 way.
  4. 7 is 6 times as likely as 2, not 3 times. Tom forgot that a double cannot be swapped, but every other pair can.

Example 3. A fair two-dice game.

Ana wins if the total is 6, 7 or 8. Ben wins on any other total. Is it fair?

  1. Ana's ways: 5 + 6 + 5 = 16.
  2. Ben's ways: 36 − 16 = 20.
  3. Not fair; Ben has 4 more ways. Even though Ana has the three best totals, Ben has eight totals.
  4. To make it fair, both need 18 ways. Give Ana 6, 7 and 8, and also 2 (1 more way) and 12 (1 more way): 16 + 2 = 18. Ben gets 3, 4, 5, 9, 10 and 11: 2 + 3 + 4 + 4 + 3 + 2 = 18. Fair.

Try it

  1. How many ways can two dice make a total of 10? List them.
  2. How many ways make a total of 6?
  3. Somebody picks horse 1. What happens to it?
  4. Which is more likely: a total of 4, or a total of 10?
  5. Which is more likely: a total of 7, or a double (both dice the same)?
  6. Kim wins on totals 2, 3, 4, 10, 11 and 12. Lee wins on 7. Nobody wins on 5, 6, 8 or 9 (roll again). Is it fair?
  7. In a race, horse 7 is in front. Your horse 9 is one square behind. Can your horse still win?
  8. Why does every row of the table have exactly one 7, but not every row has a 12?
Answers
  1. 3 ways: orange 4 and blue 6, orange 5 and blue 5, orange 6 and blue 4.
  2. 5 ways: 1 and 5, 2 and 4, 3 and 3, 4 and 2, 5 and 1.
  3. It never moves. The smallest total of two dice is 1 + 1 = 2. A total of 1 is impossible.
  4. Equally likely. Each has 3 ways. The pyramid is the same on both sides.
  5. Equally likely. 7 has 6 ways, and doubles have 6 ways (1 and 1, 2 and 2, up to 6 and 6).
  6. Not fair. Kim: 1 + 2 + 3 + 3 + 2 + 1 = 12 ways. Lee: 6 ways. Kim has twice as many.
  7. Yes. Horse 7 is more likely to move each roll, but your horse can still get lucky rolls. Likely is not certain.
  8. For 7, every orange face (1 to 6) has a partner (6 down to 1) on the blue die. For 12, orange must be 6 and blue must be 6; if orange is anything less, blue would need more than 6, which does not exist. So 12 appears only in the last row.

Common mistake

Treating "3 and 4" and "4 and 3" as the same way. They are two different ways to roll, as you see the moment you name the dice. Doubles like "3 and 3" are only one way. Forgetting this makes 7 look less likely than it is.

Thinking the most likely winner will probably win. Horse 7 is the best horse, but it loses about 2 races in 3. When there are many possible results, even the most likely one can be less likely than "something else".

Where it leads