Tossing Coins
If I toss a coin, what is the probability of it landing heads? You may say 50%.
But if I toss the coin 10 times and each time it landed tails, how would you bet the outcome of the eleventh toss?
I am not sure you will say heads or that the probability of a heads is still 50%, although you were taught that the next throw is independent of the previous ones.
But that is only true is the coin is a fair one, ie equally probable that it land on either side, which we have assumed implicitly. This may not, however, be true. It could well have been a weighted coin.
Now then does the evidence of 10 tails in a row count in any way? Does it better help me determine whether my original assumption of a fair coin was, well, fair?
If the coin have been fair then to get 10 tails is a row is an extremely rare event with a probability of (0.5)^10 or in notations can be written as:P(10 tails in a row|fair coin) = (0.5)^10 = 0.0009765625, or 0.098%On the other hand if we now entertained a slight suspicion that the coin is not fair, say with a probability of 10% that 90% of the time the coin land tails, how likely then is 10 tails in a row?
The answer is:P(10 tails in a row|coin land tails 90% of the time) = (0.9)^10 = 0.35, or 35%Evidential Reasoning
Now in the real world what I see are the evidences, eg the 10 tails in a row. What I want to know is the answer to the question is the coin a fair or weighted one based on the evidence that I see. To answer this question we use the Bayesian notion of probabilities. What it does is to quantify how probable one hypothesis over another given a set of evidence.
Notionally it is written:P(Hypothesis|Evidence) = P(Evidence|Hypothesis)xP(Hypothesis)/P(Evidence)
and
P(Evidence) = P (Evidence|Hypothesis) x P(Hypothesis) + P(Evidence|not(Hypothesis)) x P(not(Hypothesis))
More compactly it can be written as P(H|E)= 1/(1+1/SNR)
where SNR = P(E|H)P(H)/P(E|~H)P(~H)
SNR stands for signal to noise ratio.So in the given example above, which is a more likely hypotheses: is the coin weighted or not?
Lets call H1=coin is fair and H2=coin is 90% likely to land tails, and E10=the ten tails in ten toss.
We need to take one more step that is to assign prior probabilities to H1 and H2, ie how likely we find a weighted coin versus a normal coin. To do this we need to assess the context and circumstances in which we are making this assignment. If the person throwing the coin is someone we know and we are just playing some games in the bar on a beer drinking night, then the probability of H2 is that of a mismanufactured coin. On the other hand if the one throwing the coin is some who does magic as a profession, then the likelihood he has this special coin goes up.
So lets assume the context is a game amongst friends. And then the probability of this rare coin is say 0.01.
Now we can calculate.P(H1|E10) = 1/(1+ (0.35*0.01)/(0.00098 x 0.99)) = 0.22
P(H2|E10) = 1/(1+(0.00098 x 0.99)/(0.35*0.01)) = 0.78So it is almost 4 times more probable that the coin is not fair, on the basis of the evidence of the previous 10 tosses, and it may be wiser to put your money on tails for the next toss.
(You can also check the sensitivities of these results on the prior probabilities, say from 0.1 to 0.001. With increasing evidence you can see that the posterior probabilities converge regardless of what starting priors you used, ie the evidence begin to speak for itself.)
God as a Hypothesis
Now we can take the same approach in thinking about the probability of God on the basis of evidences.
We can entertain two hypotheses, namely there is God, call this hypothesis, G, and there is no God, ~G.
Now we have to assign prior probabilities for both hypotheses before evidence. If you say that the P(God) is exactly zero, then we need not go further for the question of evidence for God then is a meaningless one. But if you are still reading this sentence, then I assume you are saying that P(God) is not zero. It does not matter if P(God) is a very very small number. For as long as it is not zero it still counts as a hypothesis.
Next we have to consider what evidence are valid. Let’s call this set of evidences E={e1, e2, e3 … }
So the question is this: what is the probability for God given evidence set E? or notionally:P(G|E) = P(E|G) x P(G)/P(E)The Talking Donkey
Lets take one evidence: a talking donkey.
What is the probability that a donkey speaks, if there is God? Well if God is all powerful, surely it is very possible. So let say the probability is some number, maybe 80% or 90%.
On the other hand can a donkey ever speak if there is no God? I think it is an almost zero probability.
SoP(God|Donkey speaks) = 1/(1+ P(Donkey speaks|~God)/(P(Donkey speaks|God))xP(~God)/P(God))Now given that P(Donkey speaks|~God) is almost zero, then the whole expression in the denominator is just slightly above 1, regardless of what the values are for the other expression. So P(God) can be 1 preceeded by a milliion million zeros, the denominator is still slightly above one, which makes the entire reciprocal close to 1!
That is it does not matter if the probability of God is very very miniscule. As long as I have evidence that cannot be explained in any other way but by the existence of God, the entire probability of God becomes close to one.
And this is true for any rare or uncommon event. It is just a law of probabilities.
You may disagree on the exact numbers of the probabilities, eg that P(Donkey speaks|~God)=50%. Well you can do so, but I would have to question what is your basis for you saying so? Have you heard a donkey speak for yourself? What is the reason for thinking that it is as likely to hear a donkey speak as a coin landing heads or tails?
And further we can consider multiple evidences. I can suggest an evidence set E={donkey speaks, walking on water, resurrection of the dead, water to wine, etc etc}
You can do the calculations for P(G|E{e1,e2,e3 … }) (the notion of ‘conditional independence’ have to be introduced) but I would like to address the issue of evil.
The Evidence of Evil
How does an evil count for or against the probability of God?
Well we know for certain that evil exists, ie P(evil)=1.
Now what is the P(evil|God)? Now if you say it is zero, then P(God|evil) is zero. Now this has been a ‘standard’ argument against the existence of God. I cannot argue against you and it is true for you and your concept of God, or more precisely, your god. For this god is of your own imagination and creation, a god that is wholly good without evil. But is this who God really is?
Here I have to introduce Jesus Christ. He said he is God, and other than mad men and lunatics, I have not heard anyone else say that he is God. Let me take Jesus at face value, namely that he is God.
So evil was and is in the world, and likely ever will be. And when Jesus was here on earth, certainly evil existed: for Jesus’ crucifixion was manifest of such evil. So, P(evil|God)=1.
Now what about P(evil|~God)? Well I do not think atheist deny that there is evil. So I say P(evil|~God)=1.
So P(God|evil) = P(evil|God)x P(God)/P(evil) = P(God), ie evil does not diminish nor improve the probability of God.
Coversely P(~God|evil)=P(evil|~God)xP(~God)/P(evil)=P(~God). And this is arrived at without considering Jesus Christ, ie atheist cannot say that evil is evidence for there being no God.
Thus evil is a neutral piece of evidence: it favours neither the probability for God nor no God.