Tuesday, December 11, 2007

About determinism and tossing coins.

"Head or tail?"
I lost.

Probability science says that there is perfect uncertainity - non-determinism - in tossing a coin. And when there is perfect uncertainity, they say there is an equal probability for each one of the possible result. Since the sum must be 100%, then, there must be 50% for head and 50% for tail.
There are several consideration to say in order to parcially correct this hypotesis, for example, considering one particular coin that is going to be tossed, maybe the weight of one side is heavier than the other side, so that, when the coin is flying it likes better to fall with that side down, therefore there would be more probabilities that the result is the other side. Or we should take in consideration also the way the coin is tossed. For example i use to put the coin on the thumbnail of my right hand, and release it after pulling with the tip of my index. Some other people throw the coin some other way, and it can affect the path the coin is taking when it is flying.
Maybe we should mention also the speed and direction of the wind... And what about the Coriolis force?

The Coriolis force is something scientists use to justify strange effects that some events can have on the surface of the world. For example opening the hole of a bathtub full of water, a circular motion of water around the hole is produced, which is spinning always clockwise in the northern emisphere and counterclockwise in the southern. That is due to the fact that, as effect of the rotation of the Earth, in the northern emisphere something tied to the surface moves towards east if exposed to south faster then if exposed to north, since the circumference of the southern parallels is major of the circumference of the northern. To understand it, imagine to put a ring on the top of a ball. The most the ring is larger, the most it falls towards the "equator". If you put two rings and the ball is spinning, both of the rings make the same number of turns per minute, but the speed of a point on the side of the bigger one is faster than the speed on the side of the other. If one tries to walk straight on the north emisphere, then, he tends to turn right, or, in other words, to spin clockwise.
The force of Coriolis can influence the motion of a coin when it is flying after having been tossed. Or not?

Anyway, if we don't consider all those facts, which thing makes our life easier, we should say that since we have no idea if the coin is going to fall one or the other side, we conclude that there is equal probability for one or the other result. We don't know, so it can happen in 50% of cases. This is kind of weird!

"Composed probability" is the calculation of probability observing more than one event. Given two un-related events, the probability that one and the other happen is given by the probability that one happen multiplied by the probability that the other happen. On the opposite, the probability that one or the other happens is obviously given by 100% minus the probability that the first doesn't happen and the second doesn't either. In other words, given 1=100%, p1 the probability of event 1, p2 the one of event 2 we have:
probability of event 1 and event 2 = p1 * p2;
probability of event 1 or event 2 = 100% minus probability of not event 1 and not event 2, which is 1 - (1 - p1) * (1 - p2).
As an example, let's calculate the probability that tossing two coins we got at least one head. The cases are:
- two heads, winning
- the first head and the second tail, winning
- the first tail and the second head, winning
- two tails, loosing.
We win 3 times on four, which is 75%, infact the probability the first is not head is 50% (or 0.5), the probability that the second is not head is 50% too, so, the probability of one or the other being head is 1 - 0.5 * 0.5, which makes 1 - 0.25 = 0.75 = 75%.
Another way to consider the problem is this: we toss the first coin. If the result is head (50%), we win, and we don't need to toss the second. If the result is tail (50%) we have 50% of the remaining probabilities (which is 25% of the total) that it comes out head, so we win for the 50% + 25% = 75% of cases.
Extending the calculation to a higher number of events, the probability that atleast one happens is given by:
1 - (1 - p1) * (1 - p2) * (1 - p3) * ....
For example, tossing 10 coins, the probability that atleast one is head is given by:
1 - 0.5 * 0.5 * ... 10 times, or 1 - 0.510 = 1 - 0.0009765625 = 0.9990234375 = 99.90234375%. Almost 100%!
Going a little further, if we toss 100 coins, the probability is 99.999999999999999999999999999921%. Not bad, uh?

We said that when there is something we don't know if can happen, there is 50% of probabilities it can. The funny conclusion of this is in the following example.
Let's imagine one planet in another star system. We don't know anything about that planet except the fact that it exist. The question is: do elephants exist on that planet? we have no idea! So we must say that the probabilities are 50% (being that the other 50% are the probabilities elephants do not exist). And what about whales? Same thing, in that planet there can be whales with probability 50%. So, how many probabilities there are that elephants or whales exist in that planet? Just like tossing two coins: 75%. And what about mices? and crikets? and flees? Human beings? Not to count tomatoes or apples or bacteria of Antrax. If we take in consideration only 100 different forms of life, the probabilities that atleast one of them can be found on that planet are therefore 99.999999999999999999999999999921%.
In conclusion, if we take in consideration one planet of which we don't know anything at all, there are almost 100% of probabilities that there is atleast a form of life. Would you believe it? I wouldn't.
One explaination of this paradox is that it's not real that we don't really know nothing at all about that planet. For example we know the type of material it is made of (atoms) because we believe that the whole universe is made in that way, we know that it can be gasiform, in which case life cannot exist as we know it, or it can be big or little, depending on which force of gravity can be too strong and smashing any kind of life or too weak to keep gases attracted and so to have an atmosphere in which life can breathe. We know that, to allow life, the chemistry of that planet must allow the formation of molecula on a certain atomic model and blah blah blah...  well... i am not a scientist so clever to know all those details, but what i want to say is that it's not real that we don't know anything at all about that planet. Infact what we know is that the conditions to find a form of life on a planet are really difficult to have. We know that.
But if we didn't know it, would we conclude that there would be almost for sure some kind of life on any unknown planet?

The wrong thing is that we give to an event a lot of chances to happen even if we don't know anything at all about that event. Rather, right because we don't! If we didn't know anything about coins, would we bet that, tossing one 100 times it would result head about 50 times (whatever we mean with "about")? The fact is that we know a lot about coins. And if we don't believe, we could ourself try to toss one a big number of times and count how many times it results head. One important thing we know, for example, is that the coin has two sides, and it will fall on the table on one side or the other. This simple rule for example does not apply on a spaceship where there's no gravity, for the simple reason that there is not an "up" where to toss the coin nor a "down" where the coin is going to fall.

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