# Significance of outliers with multiple weighted coins

tl;dr: I have N weighted coins, each of which has been flipped some number of times (n_i), and for which I've measured the probability of a heads (p_i). How do I determine whether my largest p_i is significant?

Example: I have 3 coins as follows:

Coin number Number of flips p(heads)
1 100 0.72
2 200 0.51
3 50 0.50

Current thoughts:

• I think this is somewhat related to the multiple comparisons problem and family-wise error rate? But these aren't really statistical tests with a notion of significance.
• Maybe just plotting a bunch of beta functions is the best way to visualize this?

Background: I'm looking at the winrates of ~200 cards in Dominion, specifically those bought from the Black Market. Because the Black Market is randomly seeded each game, and some cards haven't been around as long, each one has a different number of occurrences. Generally, winrates for these cards vary around the 60% mark, with a few outliers in the 40s or 70s (ignore the "corrected" column for now):

• Frequentist statistical significance is largely a question of how big $n$ is. Eyeballing your table, it is clear that the coin 1's flip outcomes would very unlikely result from a fair coin (small $p$-value). Jul 13, 2023 at 19:26

For the first coin, where after 100 flips we observed 72 heads, the probability that our next flip is a head is $$\frac{72 + 1}{100 + 2} \approx 0.716$$.