Nontransitive dice
Find dice that beat one another in a cycle, then test how rolling twice changes the best response.
Count winning face pairs
1 question · No time limit
Example
Example
Which die is more likely to beat the other: A or B?
Each of the six faces is equally likely. Rolls are independent. Roll each die once and compare its total. A larger total wins; equal totals are ties.
1. Predict
Rolls per die: 1
Die A
Die B
Roll the dice to compare the results.
Make a prediction first. It is not scored.
2. Test your idea
A versus B
- Runs
- 0
- A: Wins
- 0
- B: Wins
- 0
- Ties
- 0
These are random trials, not exact probabilities.
Try the controls, then check your solution.
Why?
A versus B: 16 wins, 20 losses and 0 ties among 36 equally likely outcomes.
Compare wins with losses. Ties count as neither; they remain in the total.
A > B
Total 14 occurs in 2 ways. It beats 2 opponent outcomes: 2 × 2 = 4 wins.
Total 18 occurs in 2 ways. It beats 2 opponent outcomes: 2 × 2 = 4 wins.
Total 22 occurs in 2 ways. It beats 4 opponent outcomes: 2 × 4 = 8 wins.
Revisit your prediction
Keep exploring the experiment.
3. Change the rules
A linked challenge. This answer is not scored.
Continue your unfinished round
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Starting the new round uses one practice start.
- Answered
- 0 / 1
- Active time
- 0:00
Round complete.
Optional and off by default. Share starts, completions, replays and error categories. For visits from ChatGPT, starts and completions can also include the first QuickenBrain page and broad country. No answers, drawings or names.
How this data is usedHelp
Try the controls, then check your solution.
Your first checked solution is recorded. You can keep experimenting afterwards.
About this practice
Learn a method
Compare dice in pairsFind dice that beat one another in a cycle, then test how rolling twice changes the best response.
For a broader mix of practices, explore the probability practice test. Membership is required to start a test.
Report a problem
Optional and off by default. Share starts, completions, replays and error categories. For visits from ChatGPT, starts and completions can also include the first QuickenBrain page and broad country. No answers, drawings or names.
How this data is used