We typically say that there is a 1⁄2 chance of heads, but what we are implicitly saying is that given a probability measure P on the measurable space ({heads, tails}, {{}, {heads}, {tails}, {heads, tails}})
Giving a probability of one event only implies we think that particular event is possible. It doesn’t say anything about what other events we are considering, so there is no necessity to describe the entire space of possibilities.
Many people act as if there is some number they can assign to any event which tells them how likely it is to occur and questions of “probability spaces” never enter their minds. What does it mean that something happens 1% of the time? I don’t know; maybe that it doesn’t happen 99% of the time? How is 1% of the time measured? I don’t know; maybe one out of every 100 seconds?
Under a Bayesian interpretation of probability, which is generally used here, probability does not express how frequent something will occur. Instead, it represents your belief an event will occur or a proposition is true. Then p=0.01 means “possible enough to consider, but very doubtful”. I think most people naturally adopt a Bayesian perspective, so I’m not sure what the problem is.
Giving a probability of one event only implies we think that particular event is possible. It doesn’t say anything about what other events we are considering, so there is no necessity to describe the entire space of possibilities.
Just because you don’t care about measuring other probabilities in the space doesn’t mean that you can ignore it. If you don’t know what the space is, it’s like taking a blank piece of paper, putting an “x” on it, and saying that’s where the treasure is buried: not only do you not know the territory, but you don’t even know enough about the map for that “x” to have any value.
I think most people naturally adopt a Bayesian perspective, so I’m not sure what the problem is.
I think you’re giving too much credit here. Go out and slip into casual conversation a remark about the probability of something and see how people treat it. You could be right about the human brain, though, and maybe it’s really a First World problem created by “numerical literacy” education in schools to try to help people read the news. Every time they hear a percentage they think of the frequentist interpretation they learned in school.
Giving a probability of one event only implies we think that particular event is possible. It doesn’t say anything about what other events we are considering, so there is no necessity to describe the entire space of possibilities.
Under a Bayesian interpretation of probability, which is generally used here, probability does not express how frequent something will occur. Instead, it represents your belief an event will occur or a proposition is true. Then p=0.01 means “possible enough to consider, but very doubtful”. I think most people naturally adopt a Bayesian perspective, so I’m not sure what the problem is.
Just because you don’t care about measuring other probabilities in the space doesn’t mean that you can ignore it. If you don’t know what the space is, it’s like taking a blank piece of paper, putting an “x” on it, and saying that’s where the treasure is buried: not only do you not know the territory, but you don’t even know enough about the map for that “x” to have any value.
I think you’re giving too much credit here. Go out and slip into casual conversation a remark about the probability of something and see how people treat it. You could be right about the human brain, though, and maybe it’s really a First World problem created by “numerical literacy” education in schools to try to help people read the news. Every time they hear a percentage they think of the frequentist interpretation they learned in school.