Improbability

Through my journeys on the interwebz and my conversations with actual real life people, I have found that humans (in general) have very a deep reverence for very large numbers. The figure of speech “one in a million” is an example of that. Never mind the fact that there would be about 7,000 people on a planet of roughly 7 billion that would qualify to be “one in a million” (about 300 of those would be patriotic Americans, no doubt). The reason I point this out is to show that what we consider rare from our every day vantage point, statistically becomes an inevitability given a large of enough sample.

As an example, suppose you take a standard deck of 52 playing cards, shuffle it well, and deal yourself a five card poker hand. The chances of you getting the first card you are dealt is 1/52. The chances of you getting the second card you got is 1/51 etc. Those odds don’t seem very large. But when you look at them in combination with one another, the chances that you got the precise hand you were dealt in the order that you got them is 1/311,875,200. By most people’s standards that is a pretty rare event. Lets extrapolate that further. Suppose instead you place each card face up in turn from left to right until all 52 cards are laid out in a line. The odds that they are in the order that they are in is 1/52! (That’s 52 factorial, not “OMG 52!!!!1!!”) or 1/8.065*10⁶⁷. For clairity, it is

1/80,658,175,170,943,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Rare things happen all the time and we shouldn’t be surprised by them. If someone was told that they had a 1/8.065*10⁶⁷ chance of any specific good thing happening to them, they would probably consider that to be a miracle. I don’t however think that they would consider the specific shuffling of a deck of cards to be all that miraculous.

The reason I bring this up is because of the supposed improbability of our existence when explained by natural forces. Rarity is often conflated with impossibility in everyday life, and for the most part it works out okay. It is mental short hand that humans employ and generally works out to our benefit. We spend nearly no time on things that are statistically improbable, save for a few exceptions (notably the Lottery or any gambling). The trouble comes in when these rare things do occur with respect to the things we happen to believe are significant.

Example: Cancer cases are known to rarely have natural remissions. When looking at a large sample (with a basic understanding of biology, immunology, etc.) we would expect to see a few remissions that are not connected with treatment. However, it is very likely that any given individual that experiences a natural remission will place a very large amount of significance on that particular event. When you look at the situation from the perspective of the “one in a million”, your situation seems too improbable to happen by chance. But do not forget about the 7,000 other people feel the exact same way.

Evolution is often criticized with an appeal to large numbers (in general: humans are far too complex and improbable to have happened by chance.)  If you think of the individual minute mutations that occur on the genetic level as the individual dealings of a card from the deck, the probability of each mutation seems much more understandable. As mutations continue to occur in a species, the probability of each individual mutation does not increase. However, the probability of the cumulative mutations that you can or do get become seemingly astronomical the more you layer on those mutations. You should not be impressed by those large numbers!

When you begin to examine the sheer size of the universe and the number of things that there are in it, your concept of improbability beings to change. In a Lecture delivered by Prof. Lawrence Krauss entitled “A Universe from Nothing” (look it up on Youtube, it’s great.), Dr. Krauss says about exploding stars:

“…only one occurs every 100 years per galaxy, there are enough galaxies that if you put your hand up at night and looked in dark spot in the sky the size of a dime, with a large enough telescope you’ll see 100,000 galaxies.  And that means that even though stars explode only every 100 years in a galaxy, in a given night you’ll see 10 stars explode.”

Remember now, our own galaxy has 100 billion stars in it. 1 out of 10,000,000,000,000,000 (10 million billion) stars in a dime sized area of our sky explode every single night.