Pattern races
Race coin patterns and discover why equally likely sequences can have different waiting times.
Find a pattern in a short run
1 question · No time limit
Example
Example
Which pattern is more likely to appear at least once in 9 tosses?
H means heads and T means tails. Tosses are independent, with P(H) = 1/2. Pattern A: HTH. Pattern B: HHH. Read each pattern from left to right.
1. Predict
Pattern A
P(H) = 1/2
Pattern B
P(H) = 1/2
Heads (H) · Tails (T)
Run the experiment to inspect the coin sequence.
Make a prediction first. It is not scored.
2. Test your idea
- Runs
- 0
- Pattern A: Observed hits
- 0 / 0
- Pattern B: Observed hits
- 0 / 0
These are random trials, not exact probabilities.
Try the controls, then check your solution.
Why?
The exact chance of at least one occurrence is 39/64 for A and 119/256 for B. These count complete sequences, so an overlapping occurrence is not counted as a new independent chance.
To count exactly, keep the unfinished prefix after every toss. A completed pattern becomes an absorbing match state. Multiply H branches by the H weight and T branches by the T weight, then add branches that reach the same state.
Revisit your prediction
Keep exploring the experiment.
Change one thing
3. Change the rules
A linked challenge. This answer is not scored.
Continue your unfinished round
Your answers are saved in this tab. Continue from where you left off.
Continuing uses no additional practice start.
Discard the unfinished round?
Your answers from this round will be discarded. You can choose your settings before starting a new round.
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
Why patterns overlapRace coin patterns and discover why equally likely sequences can have different waiting times.
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