Bayesian Slots

Four machines, each with a secret payout rate your device chose. Forty pulls. Out-earn the computer.

How to play

Each of the four machines pays out with its own secret chance. You get 40 pulls.
Tap Pull on any machine. A win scores one point.
After each of your pulls the computer takes one too, using Thompson sampling. Finish with more wins than it.
Each curve is your belief about that machine's payout rate, and P(best) is the chance it is the best of the four. Try uncertain machines early, then stick with the good ones.

Playing with a class? Tap Join a class room and enter the code you were given, and your games go on the class leaderboard.

Your pull

The machines

Bayes' rule on your last pull

Prior Likelihood (rescaled to the same area) Posterior

the machine's payout rate p

You

Computer

Pulls

Wins

The computer: Thompson sampling

Before every pull it draws one plausible payout rate for each machine from its own posteriors, then pulls the machine whose draw is highest. So it pulls each machine with exactly the probability that machine is the best.

What is hidden, and how we learn it

Each machine pays out with a hidden probability p. Your device drew all four at random between 0 and 1, and told nobody.

Prior: every p from 0 to 1 equally likely, which is Beta(1, 1). Likelihood: a win has probability p, a loss 1 − p. Posterior: after w wins and l losses it is exactly Beta(1 + w, 1 + l). The Beta prior and the win/lose likelihood are conjugate: updating is just adding one to a count.

The tension: pull the machine that looks best, or try one you are unsure about in case it is better? That is the explore–exploit trade-off, and the width of each posterior is what settles it.