Part of this is simply about having more information to allow us to make better judgments. If I ask "what's the probability that it is going to rain on Sunday" my answer will be different if I have no other information than if I know we are talking about a particular place and time of year. But then we can represent this as different probabilities: P(rain on Sunday) vs. P(rain on Sunday | we are in Seattle in October and are having a rainy season and it's been overcast and rainy for the last 3 days and is overcast today and the weather app said it was going to rain). And note here that the conditional probability is different from the joint probability! (See conjunction fallacy below.) In the conditional case we are saying that those other events have already happened; in the joint case, we are asking about the probability of it raining AND those other events happening.

This is also related to how we define a probability space. Once we agree on the possible events and outcomes, we can go about assigning or calculating probabilities to them. This is something that we don't always do exactly the same way, especially in ambiguous circumstances. For example, you can interpret my rain question as a question about it raining or not raining on Sunday (those are the possible events that make up our probability space) or as a question about the probability that it will rain on that day of the week vs. other days of the week. We can be primed to think about the problem in either way and this can bias our answers (Criag and Rottenstreich 2003 <- pdf!).

Which bringd up the issue of language / assumptions. When someone asks "what is the probability of rolling a 3" they are leaving out "on one roll of a fair, 6-sided die that has a different number on each side". We generally understand what poaaible outcomes we are considering (the probability space) when we are asking these questions, but sometimes need to be more specific.

Part of what you describe might also be related to the conjunction fallacy which is an example of a kind of probability error people tend to make in judging two events (A and B) to be more likely than just one of those events (A). The classic example is the "Linda problem" from Kahneman and Tversky (1981) that I copy in full from wiki:

> Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

> Which is more probable?

> Linda is a bank teller.

> Linda is a bank teller and is active in the feminist movement.

> The majority of those asked chose option 2. However, the probability of two events occurring together (that is, in conjunction) is always less than or equal to the probability of either one occurring alone—formally, for two events A and B this inequality could be written as Pr(A and B) <= Pr(A) and Pr(A and B) <= Pr(B).

> For example, even choosing a very low probability of Linda's being a bank teller, say Pr(Linda is a bank teller) = 0.05 and a high probability that she would be a feminist, say Pr(Linda is a feminist) = 0.95, then, assuming these two facts are independent of each other, Pr(Linda is a bank teller and Linda is a feminist) = 0.05 × 0.95 or 0.0475, lower than Pr(Linda is a bank teller).

So I suppose the answer to your question is that it depends on the context and what you are actually interested in estimating. For that we need to be clear and precise and sometimes say more than just one sentence that can be interpreted in many ways.

A similar discussion emerged after Mexico has been hit by an earthquake three times on the same date. It is nicely discussed in this podcast: https://www.bbc.co.uk/programmes/m001cq31 In essence, if you only wonder how likely an event was after it happened, it is incorrect to look at the probability of this happening to a specific person at a specific time and place. It could have happened to another person, at a different time and at a different place and you would still have read about it and wondered about how unlikely this event seems.

Good question! To answer it, I'll pose another one: why don't we take it as evidence the lottery was rigged every single time someone wins (i.e. not twice in a row but just once)? In other words why is: Given the lottery isn't rigged, the probability that SOMEONE will win is (close to) 1 the relevant statement, instead of: Given the lottery isn't rigged, the probability that JULIET will win is very small. Clearly if we use the latter approach, we will (almost) always find "evidence" the lottery is rigged even when it isn't, so this can't be correct. And the reason why is that we've formulated the probability statement AFTER we know Juliet won.

So, with someone winning twice in a row; the relevant statement is: given the lottery isn't rigged, what is the probability that SOMEONE will win twice in a row? Assuming that's reasonably high, we can't use this as evidence the lottery is rigged. But let's assume it's low; you may rightly ask: why isn't the relevant question: given the lottery isn't rigged, what is the probability that someone will win twice (at any two time points)? Why are we assigning special weight to the fact that it's twice in a row? And here is where there is really no good answer. Yes it's true we formulated the former probability statement before knowing someone actually did win twice in a row. Or did we? What if Juliet had not won back-to-back, but twice within the space of a week? Would we still insist this was evidence of the lottery being rigged? These are the questions that keep statisticians awake at night.

The results of two completely separate lottery events do not affect one another. It’s the same idea as flipping a coin twice- just because one toss came up heads this has no effect on the outcome of the second toss.

So really the relevant question here is ‘what is the probability of someone playing two state lotteries at once’. The probability of this is high- people who gamble on terrible odds are likely to do so in multiplicity.

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Let's say you let J be the event that Juliet wins two state lotteries back to back, and let A be the probability that anyone wins two state lotteries back to back. Let R be the event that the lottery is rigged. What you really want to know is the posterior probability that the lottery is rigged, and this could be P(R|J) or P(R|A).

I'm interpreting your question as asking: how should I choose whether to care about P(R|J) or P(R|A)?

Fortunately, if we know nothing about Juliet, it doesn't really matter. See, you can calculate P(R|J) = P(J|R)P(R)/P(J), and you can also calculate P(R|A) = P(A|R)P(R)/P(A). I claim these calculations will produce very similar values when we know nothing about Juliet. P(R) is of course common to both. Canceling it from both expressions leaves P(J|R)/P(J) in the first expression and P(A|R)/P(A) in the second.

In the second expression, both the numerator and denominator are larger. There are at least a thousand people who play the New Jersey lottery^([citation needed]), so we expect P(A) > 1000P(J). But similarly, if the lottery is rigged, it could be rigged in favor of any of those one thousand people, so we also expect P(A|R) > 1000P(J|R). If we genuinely do not think any of those people is more likely to be favored by the rigging than any other, then the ratios will be exactly the same and our calculated probabilities will be the same too. In this case it doesn't matter whether we care about P(R|J) or P(R|A).

Remember, this only works if Juliet is an arbitrary person. If we do know that Juliet is the daughter of a crooked New Jersey politician, then maybe we think that if the lottery is rigged, it's quite likely to be rigged in her favor. In that case, we might say P(A|R) = 10P(J|R), while still believing that the lottery is probably not rigged and P(A) = 1000P(J). Then P(R|J) is going to be 100 times bigger than P(R|A). In this case it would be wrong to use P(R|A), because it's throwing away important information (Juliet's crooked connections).

In summary, in both cases we should really condition on the more specific event (i.e. we care about P(R|J)), because that takes into account all the available information. But luckily for our sanity, when we have no special information about Juliet, P(R|J) = P(R|A). So even though you want to condition on all the available information, it's fine to ignore information that means nothing to you.