Probability Theory

[dick roose]

1. The problem statement:

 

A - something that is a known possibility

C - correct conditions conducive to produce A

T - many trillion years in which B proceeds

 

 

2. My question:

 

Is it true that with the above factors the probability of "A" is "1", or very close to 1?

 

3. The attempt at a solution:

 

This is what I believe Professor Steven Hawkins said about a meteor or comet striking Earth.

[DJBruce]

The conditions need for A would already be taken into account of for the value of P(A) so once the value is established it will not change unless the conditions itself change. If event A has a probability of occurring of P(A), P(A) will not change regardless of the number of trials (years) occur. The number of times A occurs, the expected value of A E(A), would change with the number of trials. For example imagine that the probablity of getting a heads on a fair coin is P(H)=.5. The expected value of an event is the probability of the event occurring, P(H) in this case, times the number of trials, n. So lets see how the expected value changes as we vary n

 

[math] n=2, \left(.5\right)\left(2\right)=1[/math]

[math] n=10, \left(.5\right)\left(10\right)=5[/math]

[math] n=1000, \left(.5\right)\left(1000\right)=500[/math]

 

So as the number of trials increases the number of times we expect the event to occur increases. No lets consider something that is very rare like Earth being struck by a comet. I am going to make the probability of Earth being struck by a large comet in a day [math]P\left(C\right)=10^{-11}[/math]. So the probability is very low. However lets see what the expected value of is for large comets striking the Earth. The Earth is [math]4.45*10^{9}[/math] years old, and there are 365.25 days in a year. This means there have been a [math]1.66*10^{12}[/math] days on Earth. So the expected number of large comets to have stricken the Earth is:

 

[math]P\left(C\right)*n=\left(10^{-11}\right)\left(1.66*10^{12}\right)=16.6[/math]

 

So in this situation we expect that the Earth has been hit by about 17 large comets in its life. So even though the probability of the event is small with enough trials the event can occur quite a few times.

 

 

So no the probability of the event would not change to near 1 or 1. It is just that the number of trials make the event occurring eventually likely.

Please Note: The probability of a comet striking Earth is a completely made up number.

[cypress]

 

 

Is it true that with the above factors the probability of "A" is "1", or very close to 1?

 

 

It depends greatly on the probability density of A in the set of all possible outcomes that meet the conditions required for A to occur, and the probabilistic resources available to act. If the probability density of outcome A is low relative to the resources then the overall probability will remain low.

 

I suppose this might be what you mean when you say the conditions are favorable for A to occur, that is that the probability density is large compared to the opportunities for A to occur.

[CharonY]

I am sorry, but this is just gibberish. A probability density function describes the likelihood of a random variable to fall within a given interval. An example would be, say, having a size distribution of kids in class and calculate the probability of a kid being between e.g. 175 and 180 cm.

Note, that the larger the interval, the close we approach 1.

The described example is much simpler and can be described with a uniform function and taking f(x)=const. In that case the probability would simply be described as by DJBruce (or simply the integral over the total time under consideration).

Edited August 24, 2010 by CharonY

[cypress]

DJBruce described a simple form of the expectation value, not the probability.....

 

In his example, the probability of A would of course be close to one, but in the general case it would in fact depend on the probability function, the resulting probability density for the events that would result in outcome A and the opportunities or resources available to cause events that make of the set of events that A is contained within. In DJBruce's example combines all the events that lead to outcome A along with the individual probabilities into an overall probablity P(A) which is fine. In the numerical example though it is expressed as a probability density in the form of probability per day. Then in the example, the probabilistic resources (the opportunities for A to occur) are the number of days for which the events are in play. To obtain the overall probability of a comet striking the earth, we could use this formula, (1-(1-P[C])^days)

[DJBruce]

DJBruce described a simple form of the expectation value, not the probability.....

 

In his example, the probability of A would of course be close to one,

 

No P(A) in most cases would not be any where near 1.

 

 

but in the general case it would in fact depend on the probability function,

 

What is the "general case"?

 

the resulting probability density for the events that would result in outcome A and the opportunities or resources available to cause events that make of the set of events that A is contained within.

 

As CharonY explained your use of the term probability density is incorrect.

 

 

In DJBruce's example combines all the events that lead to outcome A along with the individual probabilities into an overall probablity P(A) which is fine.

 

No it doesn't. My entire point was that even with the fact that P(A) is incredibly small, with a large enough number of trials it will happen. In no point do I create an overall value P(A).

[CharonY]

The question here is really what probability is to be calculated. At the moment we are being sloppy. Let us assume that the question is how likely is an event to happen at least once after n repetitions with a given p for each indivudal trial. The probability of the event to occur at least once is simply 1- probability of it never occuring.

Not that using that equation we do not use the probability mass (not density as it is binomial distrubution we are dealing with) function to solve it. I.e. the equation is only valid if the individual probabilty p is used, not P(x). However, if desired it can be calculated using 1-P(0), though it would require unnecessary additional operations (and result in the same equation). Moreover calling n a probabilistic resource is extremely odd and certainly not mainstream.

 

I now suspect that the overall confusion is the result of incorrectly applying terms. I will try to clarify. If we denote the hit with a comet as success and take the number of hits after n trials as outcome, then the probabilty mass function indicates the probabilty of having precisely X successes in n trials. So we would use it, if we want to know the probablilty that in 5 million years (if we take each day, or month for example as a trial) the earth was hit precisely 4 times. The probability of each individual event does not change, but the probability of the number of successes is of course different.

 

Note that we could also turn the question around and ask how many trials does it take that the even occurs the first time. We would then use a geometric probabilty mass function with X now being the number of trials. Using its cumulative distribution function, we could determine the probability of the even occuring after X trials.

[dick roose]

Hi All,

 

I appreciate all the replies, but I am not sure my question has been answered. DJBruce posted "My entire point was that even with the fact that P(A) is incredibly small, with a large enough number of trials it will happen." May I take this as a "yes" the probability in this case is "1"" answer. I have quite a bit riding on this being true. Can anyone be more specific? What follows is my take on the subject!

 

The probability that life would occur somewhere in the universe being "1" is a problem which I have not yet resolved to anyone's satisfaction. The idea occurred to me as I was watching Steven Hawkins on a PBS special and he said that given enough time the probability of Earth being struck by something from outer space was "1". To my way of thinking there are many more chances for life to erupt from inanimate elements than there are things in the space surrounding our Sun. We are not talking infinities; we are talking finite numbers. For instance the number of elements which must come together in the right sequence and in the right proportions is about 12. Although modern life is composed of 26 elements, it has been postulated that a primitive life form could probably get by with 12 elements, and possible only 6. I do not know where I got that information, probably out of Science News or one of the other science magazines I used to read. At one time I had two boxes full of articles for my book but a fire got them.

 

If we consider the case of 12 elements then the number of combinations need to test all combinations is 12 factorial or 479,001,600. If we give the proportions a range of 100 to 1, then we have a possible combination count of 12! X 100! which is in the range of 10^31. Howsoever, if we examine the possible experiments that can be conducted in a universe like ours, and the experiment needs to work only once, we are looking at a much larger number. I have considered one micro liter of water to be sufficient to mix and combine all of the elements required to produce a single living cell. The oceans of Earth are saturated with all 92 elements and they are always moving, joshing the contents around continually. If we allow 1 second per experiment we have 1440 experiments a day times 1.347 x 10^27 (number of micro liters of water on Earth) we get 1.939 x 19^30 experiments per day on planet Earth. In one year we get 7.077 x 10^35. Now if we consider only 1 star in 1000 has liquid water, then we have 7 x 10^19 planets which could possibly produce life. We now have 49.077 x 10^54 experiments per year in the universe. Even if only 1 out of 1000 experiments provided a unique sequence and even if we take a full liter of water for each experiment, we still have 49.077 x 10^45 experiments which is 14 orders of magnitude greater than the number of all the possible combinations of 12 elements in a proportion range of 1 to 100.

 

If my numbers are anywhere close to accurate then we have many, many more different experiments performed each year that there are possible combinations of 12 elements with a 100 to 1 amount range.

 

Since all combinations of the necessary elements in a range of proportions must occur somewhere at some time in the universe the probability of life spontaneously forming is "ONE" or inevitable. We humans just won the biggest lottery of them all!

 

I am sure you will let me know if my numbers are wacky. At age 74 I make a lot of math errors!

Edited September 1, 2010 by dick roose