Run Thompson Sampling by Hand on a Beta-Bernoulli Bandit

Two homepage banners, Sale and NewArrivals, start with a Beta(1,1) prior on click-through rate. Over one day of traffic: Sale gets 12 clicks out of 60 impressions; NewArrivals gets 3 clicks out of 10 impressions.

1. Compute the posterior distribution for each banner after this day's data, and report the posterior mean for each.
2. Without drawing an actual random sample, explain qualitatively why NewArrivals, despite fewer total clicks, has a real chance of being selected on the next round even though its posterior mean is lower than Sale's (assume it currently is — check this first).
3. A colleague suggests initializing both arms with a strong prior, Beta(50, 50), "to avoid wild early swings." What is the tradeoff of doing this?