Thompson Sampling
Thompson sampling draws a plausible success rate for each arm from its posterior and plays the largest draw. Uncertain arms get tried in proportion to how likely they are to be best.
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Beta Distribution
Beta(0.8, 0.2), Beta(4, 1) and Beta(400, 100) all have mean 0.8000. P(0.7 < p < 0.9) is 0.1512, 0.4160 and 1.0000.
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Prior Belief (or Prior Probability)
A prior specifies assumptions before the analyzed data. Its concentration, parameterization, predictive implications, and sensitivity all affect the posterior.