Let's consider two scenarios: 1) Somebody starts flipping a coin they claim is fair. You're not sure whether or not that's true. It comes up with 10 heads in a

Let's consider two scenarios: 1) Somebody starts flipping a coin they claim is fair. You're not sure whether or not that's true. It comes up with 10 heads in a row. 2) Somebody tells you there's a 1 in 10,000 chance of something happening each day. You're not sure whether or not this is true. It happens tomorrow. Which, if either, do you think has reasonable evidence to begin to doubt the claimed odds are the real odds? Which do you think is better evidence of the proposed claim being false? With the logic you expressed in your post you might suggest the first is better evidence of a false claim, but it's not. The odds a fair coin going 10 heads in a row is 1 in 2^10, or 1 in 1024. Even though it's 10 'events', it's ultimately just a 1 in 1024 happening if we assume a fair coin. By contrast the second event is about an order of magnitude less likely to have happened by chance. So even though the second event is only one happening, it's more likely that that event is not a 1 in 10,000 event than it is that the coin is not fair.