- These three guys go to a hotel (back in the old days) and rent a room. The desk clerk charges them $10 apiece. Afterwards, the clerk decides that he has charged too much and gives the bellhop $5 to return to the guests. The bellhop, being unusually savvy, realizes that the $5 will not divide evenly and decides to solve the problem by keeping two dollars for himself. Which is what he does. He gives each guest $1, which means that they each have paid $9 for the room, or $27 altogether.
- But wait a minute, the bellhop kept $2 and the guests paid $27, making a grand total of $29. What happened to the other dollar?

 

 

VIII. Immortality:

Rejecting the Null Hypothesis of Personal Mortality
(A Road Less Traveled)
3/20/10

 

Scene 1:

Say that you find a deck of cards in the closet and decide to play some solitaire or something.

You sit down at the table and turn over the first card. It's an ace of spades. You place the ace back in the deck, shuffle the cards and once again, turn over the first card. This time, it's the ace of diamonds. Hmm. So, you try the same thing again. This time, you get the ace of spades again.

'Wait a minute…' You do it one more time, and this time, you get the ace of hearts.

If you’re paying attention, you’re growing suspicious about this deck you found in the closet. You’re starting to suspect that you don’t have the ordinary deck that you had assumed. But, why is that? Why are you suspicious?

You’re suspicious because the probability of drawing that 'hand' is so small if the deck is a normal deck.

Let’s try that again. But, this time, the first card you draw is a 3 of diamonds, the second is a
Jack of spades, the third is a 9 of clubs and the fourth is a 9 of hearts. In this case, you probably are not suspicious.

But, of course you realize that the prrobability of drawing that hand, given a normal deck, is just as small as the probability of drawing that previous hand…

So, what’s the problem here? Why are you not suspicious of this deck, when you were suspicious of the first one?

It turns out that there are two factors causing you to be suspicious of that first deck -- and one is missing in regard to the second deck. There is nothing about the second hand that sets it apart in such a way as to suggest another plausible hypothesis… If there were, you’d be suspicious of that second deck as well. It’s as simple as that…

If, for instance, you were to remember that a few years back you had assembled a deck of 3’s, jacks, and 9’s (for some weird reason), the second hand would also stand out and would also make you suspicious. And, again, your suspicion would be mathematically justified…

Any time that new information you receive causes the overall information you have to logically favor a new ‘worldview,’ you are justified in revising your expectations accordingly. Who could argue with that?

And, that’s what happened when you drew the 4 aces, but did not happen when you drew the other hand.

The nature of a random "sample" has mathematical (probabalistic) implications about the
population the sample was taken from. The 4 aces, by being so improbable under your preexisting worldview regarding the deck, and by being so probable under a different, plausible, worldview about the deck, caused the overall information you had to logically favor the new worldview.

The "nondescript" hand, by not being especially probable under any other plausible worldview about the deck (that we can think of), even though it was specifically so improbable under the old worldview, does not cause the overall information you have to logically favor a new worldview.

By not suggesting another worldview, and thereby not setting itself apart from most other
possible hands, the nondescript hand "blends" with most other possible hands and effectively takes on the combined probability of most hands…

So as to better understand the math for this kind of situation, let’s try something where the
numbers are easier to nail down.

Say that you have 50 decks of freshly shuffled playing cards on the table in front of you. You
know that 49 of these are normal decks, but that the other deck has only aces. You don’t know which deck is the aces deck. You randomly select a deck. At this point, the probability that you selected the aces deck is 1 in 50.

Now you select the top card from the deck you chose and it’s a 3 of hearts. With that new
information, your probabilities have changed. The probability that the deck you chose was all aces has just dropped to zero.

Backing up, say that instead of getting the 3 of hearts, you got an ace of hearts. The probability that you chose the aces deck just got a nice boost. Specifically, the probability is now 1/50 times 1, divided by 49/50 times 1/13, or a little better than 1to 4, or 25%.

The logic here is that the probability of drawing an ace if you had chosen the ace deck is 1, so the “composite” probability of drawing the ace deck and then drawing an ace from it is 1/50 times 1, or 2%. The probability of drawing an ace if you had chosen a normal deck is 1/13, so the “composite” probability of drawing a normal deck and then drawing an ace from it is 49/50 times 1/13, or .075385.

So, once you draw the ace, to determine what the probability is that you drew from the ace
deck, you need to compare the two composite probabilities, and ultimately you end up being about 4 times as likely -- .075385/.02 -- to have drawn from the normal deck…

One more detour before we get to the good stuff.

In the behavioral sciences, our research takes on meaning through statistics. Say we’re trying to compare one method of teaching to another. We find that the mean (average) test score received by students being taught by method A is 5 points higher than that received by students being taught by method B. We want to know if this difference is “significant” -- and indicates that A is a better method than is B -- or is small enough to fit within the range we would normally expect by chance.

To do that, we select an arbitrary probability – usually .01 – and figure that if the probability of getting the results we did, under the hypothesis that neither method was better than the other, is less than .01, we are mathematically justified in “rejecting” that hypothesis. That particular hypothesis is called the “null hypothesis,” and we say that we can reject the null hypothesis at the .01 significance level.

We can do something analogous with the 50 decks of cards above. Say we continue where we left off above and draw another ace from the same deck. At this point, the probability is in favor of the aces deck -- but is it sufficiently in favor of the aces deck in order to "reject the normal deck hypothesis"?

No. With 2 straight aces, the probability that you selected a normal deck becomes 49/50 times 1/13 times 1/13 divided by 1/50, or about .29. Again, we're looking for .01 or less, so we can't reject the normal deck hypothesis just yet. Keep in mind that the .01 significance level is intended to be very conservative and if you had to BET one way or the other, you'd be significantly better off betting against the normal deck in such a case. You'd win almost 4 times more than you'd lose.

In order to reject the normal hypothesis in this case, under the typical behavioral science standard, we'd need to draw 2 more times and get aces each time. That would give us 49/50 times 1/13 times 1/13 times 1/13 times 1/13 divided by 1/50, or about .0017. If we bet against the normal deck every time we got these results, on the average, we'd win 998 times out of every 1000 times we bet.

I'm sorry, but I lied before. We have one more, short, detour. I haven't shown how the 5 points better average of Teaching Method A, above, suggests another plausible hypothesis. But that's obvious – which is why you won’t get anywhere asking most statisticians about this second factor. In our work, the other plausible hypothesis is always a given. The other plausible hypothesis suggested by our results is that Method A is better than method B.

You might want to take a break now, but otherwise, I think you’re ready for the big time…

According to the typical non-religious hypothesis of personal existence, YOU are a random and one short-lived accident (at most), and would never exist if your parents had never met. The same would be true if your grandparents had never met – on either side of the family. This can be traced back for … a long time. And if just one of these chance meetings had not taken place, you would never exist. It gets worse. Not only did your parents have to meet, the right sperm cell had to meet the right ovum. Otherwise, the results wouldn’t be you – it would be your brother or sister. And, as it turns out, your father probably produced quadrillion sperm cells in his lifetime and your mother was born with several hundred ova. You happen to be the specific combination of just one of those sperm cells and just one of those ova – no other combination would do. And, just think of all those potential offspring from your Dad and Cleopatra -- they never had a chance!

And worse. Since you will only live for about 100 years, there is another, infinitesimally small,
probability that has to be factored in – the probability that now would coincide with your existence. It is much more likely that now would be some other time altogether, along this infinite continuum of time…

In other words, given the typical non-religious hypothesis of personal existence, the probability of your current existence is incredibly small. So, the fact that you exist casts serious doubt upon the typical non-religious hypothesis…

Or, does it? Well, is YOUR current existence analogous to the 4 aces, or is it analogous to the 3, jack and 9s? That’s the question. Is there, or is there not, a dollar missing?

But then, it would appear to be an easy question to answer – much easier than, “What happened to the other dollar?” Here, you don’t need to ‘feel’ the appropriate mathematical logic – you have a rule to which to refer back.

First, does your current existence suggest another plausible hypothesis? I don’t know about you, but I can think of at least four somewhat plausible alternatives. Each requires a little explanation, but each is, indeed, somewhat plausible. 1) Our understanding of time, and our existence in it, is all screwed up. 2) You are ‘basic’ – a basic part of reality. 3) Life is analogous to a dream. 4) You were, somehow, intentionally created. And, it’s their total probability (at least) that counts -- not just one of their probabilities.

We’re not quite out of the woods yet, however. We’ve got to do the math. Does the OVERALL probability of one of these new hypotheses being true outweigh the OVERALL probability of the old, typical, NR hypothesis being true?

Because the probability of your specific existence is so incredibly small under the non-religious hypothesis, the overall probability of your new hypothesis (and remember, we can add the three probabilities together) need not be much at all in order to validly reject the non-religious hypothesis.

While the probabilities to be added and multiplied are amorphous at best, in search of a more
concrete understanding, we are justified in suggesting specific values towards the morphous ranges and apply the appropriate math.

Giving the NR hypothesis the benefit of the doubt on both of the following counts, we might
suggest that its ‘background’ probability (analogous to the probability of a normal deck before you turned over the first card) is 9/10, whereas the probability of you existing given the NR hypothesis (analogous to the probability of drawing an ace from a normal deck) is 1/1,000,000. And so, its overall probability given the new info of your existence is therefore 9/10,000,000. Let’s place the ‘background’ probability of one of the other hypotheses being true at 1/100 and the probability of you existing given one of the three other hypothesis at 9/10. If these are somewhat accurate, their overall probability comes out to be 9/1000! Comparing the two, we get .0000000001!

So, even being grossly conservative, the overall probability of one of the alternative hypotheses being true is MUCH greater than the overall probability of the typical non-religious hypothesis being true, and we are mathematically justified in rejecting the latter…
In other words, you’ve got GOOD REASON to believe that you are either not accidental or you are not temporary – at least, one or the other. But also, if you look closely at what we’ve done here, you’ll see that you actually have good reason to believe that you are NEITHER -- and that both appearances that you ARE, are illusions… How’s that?!

Though the typical NR hypothesis is probably the road more traveled by the intelligentsia these days, it doesn’t really make sense, is analogous to thinking that a dollar was missing, and can be mathematically rejected. Hallelujah! Scene2

 

The Math