**Rare events**

Suppose you drew 5 cards and they were all hearts. You might ask “what’s the probability of drawing 5 hearts at random from a 52 card deck?” Well, sure, that’s easy enough to calculate. The probability is 0.000495. Wow! Such a rare event! Must be your lucky day.

But… the reason you asked about 5 hearts is because that’s what you experienced. You peeked at the data, and then asked your question. I am sure if you would have gotten 5 clubs you would have asked about that. Or if you had gotten a straight.

So, one way to rephrase the question is as follows: “what’s the probability of drawing 5 cards at random from a 52 card deck that, upon viewing these cards, would have gotten my attention and prompted me to ask a question about probability?” I’m confident that any flush or straight would have gotten your attention, and four of a kind as well. So let’s stick with those. The probability of drawing either a flush, a straight or four of a kind is 0.006. While this is still a very rare event, it’s about 10 times higher than that of getting 5 hearts.

**My existence**

I have heard it argued that

the probability that you are aware right now, when your existence could have ended billions of years ago, or could have come into being billions of years in the future – this probability is so small, so insignificant, that it is practically non-existent… this could only mean the existence of God.

However, the probability calculation is incorrect. Let’s define the event A as follows:

A: I exist now out of all of the possible times I could have existed

The argument is that P(A) is essentially 0. However, the argument ignores the fact that I already do exist right now, which is why I am asking the question. I am asking a question based on data that I have already seen. We have to condition on that data. Therefore, let’s define the event B as:

B: I exist right now

What we are interested in is not P(A), but P(A|B). We have to condition on B, because B is the reason we are asking the question. It’s the data we peeked at. Well, it turns out that P(A|B)=1. Thus, it’s not a rare event and certainly cannot be an argument in favor of any religious beliefs.

**Self-indication assumption
**

The self-indication assumption (SIA) is

Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

Katja Grace presents a simple example of SIA:

For instance if you were born of an experiment where the flip of a fair coin determined whether one (tails) or two (heads) people were created, and all you know is that and that you exist, SIA says heads was twice as likely as tails.

For simplicity, let’s assume this action only happened once (one coin flip). Thus, there is only one or two people in the world, depending on whether it was tails or heads. Let’s assume if we are in two person world, we don’t see the other person.

Let’s define the event A as:

event A: the coin came up tails (i.e., the one person world)

SIA reasoning is that since there were 3 possible people including myself, and I was selected, the probability that I’m on 1 person world is 1/3.

The flaw here is the focus on me existing. I already exist, so it’s cheating to write questions about me existing after seeing that I exist. Like the card player who formed the hypothesis after seeing the cards, we’re asking the wrong question.

Instead, let’s define the event B as:

event B: at least one person exists

Event B is what we really want to condition on. We want to condition on an arbitrary person existing — there is nothing special about me in this scenario (unless you cheat and use the data you peeked at). Well, in either of these two worlds (heads or tails) there will exist someone that is wondering which world they are in. So, the fact that there is someone wondering which world they are in tells us no information about which world we are in. That is, P(A|B)=P(A)=1/2.

So, in my opinion SIA is wrong. The fact that I exist tells me nothing about the number of observers.

**Doomsday argument
**

The doomsday argument seems flawed for the same reason. It basically says that the fact that you exist is evidence that the race will die out soon.

It’s correct that if you could randomly draw a human from the N that will exist, you will probably pick one that is towards the tail of the distribution (when the population is greatest). However, we cannot think of ourselves as a random draw. It’s peeking at the data. We already exist.

It should be obvious the argument is flawed based on the fact it always comes to the same conclusion. Suppose, for example, the total number of humans to ever exist (past and future) will total N. Every human, numbers 1, 2, … , N, will at some point exist, and could wonder if humans will face extinction soon. So, if we condition on the fact that right now there is at least one human asking that question, we have no information about whether that human is close to number N. All information about possible extinction would have to rely on other sources.

on April 10, 2010 at 8:18 am |Katja GraceInteresting post.

If you found yourself wondering about any noticable group of cards, it would only be because some noticable group came up – shouldn’t you find it unsurprising even that a noticable group came up by your reasoning?

You say there is a problem with asking something about your own existence when you know that you exist, and suggest instead asking about the probability that a person would exist. You already know that a person would exist too though – how does that help?

I disagree that there is necessarily a problem with peeking at data before asking questions – there can be a problem if you take the data into account implicitly in choosing priors for your question, then use the data again to update, but I think that’s not what you are talking about. I agree that you shouldn’t be surprised by getting any particular unlikely set of cards, but I think perhaps that’s because you knew you would get something that unlikely to begin with. If you knew the pack was full of aces of hearts, plus one eight of spades, and you got the eight of spades I think you could more legitimately be surprised, even if you only wondered about it upon it happening.

The 1/2 position commits another offense after peeking at the data I think – it assumes you would certainly exist because you see that you do. To say that either world was just as likely to give rise to you is to say that the other person in the two person world couldn’t have been in the one person world – you are treating yourself as special.

on April 11, 2010 at 3:15 am |jasonKatja,

I wrote a more detailed post explaining my view of the SIA assumption (newest blog post). Hopefully that will clarify some things.

In the cards example, yes, it’s surprising that some noticeable group came up. I was just pointing out that the question you are interested in might really be “how lucky was I (i.e., how unlikely was a hand that was this good or better)?” as opposed to “how rare is this particular hand?”

Jason

on November 26, 2011 at 2:18 pm |The Umbrella Man « N=1[…] have made this point before, but this example is better than the ones I came up with. Share […]

on November 26, 2011 at 2:18 pm |The Umbrella Man « N=1[…] have made this point before, but this example is better than the ones I came up with. GA_googleAddAttr("AdOpt", "1"); […]