Probability about Jesus (Christ) existence on earth

[timhendrix]

Well where this would leave us is:

P(Jesus existed) > P(At least one of s1, ... s15 is NOT iid) = 1-P(all of s1, ... s15 ARE iid)

which is equivalent to:

P(Jesus existed) > 1-P(s1 is iid & s2 is iid & ... & s15 is iid)

and where we know

P(s1 is iid) = 28%
P(s2 is iid) = 28%
...
P(s15 is iid) = 28%

The problem is you can't compute P(s1 is iid & s2 is iid & ... & s15 is iid) only from the above list of 15 probabilities. I know that you are assuming that

P(s1 is iid & s2 is iid & ... & s15 is iid) = P(s1 is iid)P(s2 is iid) x ... xP(s15 is iid) = 0.28^15

And in many cases that is true (which is why you can find the formula on the internet). However, there is simply no rule in probability theory that allows you to do it in general. It is like saying: I know that X + Y = 10, what is X?. You simply don't have enough information to figure it out.
That's why I started out by saying your result could be true, but it could also be false, and all we can do is to state under which conditions it is true or not. Is that making any sense?

I don't know how much of that you can accept and what parts you might disagree with. Would it help if I gave a horse-free scenario under which it could be the case that

P(s1 is iid) = 28%
P(s2 is iid) = 28%
...
P(s15 is iid) = 28%

but

P(s1 is iid & s2 is iid & ... & s15 is iid)

was NOT equal to P(s1 is iid)P(s2 is iid) x ... xP(s15 is iid) = 0.28^15?

Otherwise, what could potentially convince you what I said above was true?

[Bernard Muller]

to timhendrix
I am utterly confused about your math and I am afraid you consider the 28% as implying Jesus was NOT historical. But this 28% are about NOT implying Jesus was historical because of the possibility of interpolation, (other) interpretation or dependence (on gospels). That's very different.
Maybe you should limit yourself to only 3 instead of 15 points in order to explain (in details) your thinking and calculation. That would be more understandable this way.

P(s1 is iid)P(s2 is iid) x ... xP(s15 is iid) = 0.28^15 = 0
So what would be the numerical result of P(s1 is iid & s2 is iid & ... & s15 is iid)? With your "&", are you meaning: probability that all the 15 points are iid?
Why should these two equations provide the same numerical result?


Cordially, Bernard

[timhendrix]

to timhendrix
I am utterly confused about your math and I am afraid you consider the 28% as implying Jesus was NOT historical. But this 28% are about NOT implying Jesus was historical because of the possibility of interpolation, (other) interpretation or dependence (on gospels). That's very different.
Maybe you should limit yourself to only 3 instead of 15 points in order to explain (in details) your thinking and calculation. That would be more understandable this way.

P(s1 is iid)P(s2 is iid) x ... xP(s15 is iid) = 0.28^15 = 0
So what would be the numerical result of P(s1 is iid & s2 is iid & ... & s15 is iid)? With your "&", are you meaning: probability that all the 15 points are iid?
Why should these two equations provide the same numerical result?


Cordially, Bernard

Hi,

I will get to the numerical results in a moment, but before that I think it is good that we agree on how far probability theory CAN take us, and then later discuss where we might disagree -- keep in mind that as far as I know what I do below is similar to the steps you take as far as I can tell.
As I understand you you agreed with my post before my previous post and so that:


P(Jesus existed) > P(At least one of s1, s2, s3 is NOT IID)

(think about what that means). The probability that AT LEAST one of s1, s2, s3 is NOT IID is the same as 1 minus the probability they ALL are IID (because the probabilities have to add up to 1). So we get:

P(At least one of s1, s2, s3 is NOT IID) = 1 - P(s1 is iid AND s2 is iid AND s3 is IID).

Which gives us (I use AND here instead of &)

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is IID)

Does all that seem okay with you?.
In addition to this, we started out by assuming that

P(s1 is NOT iid) = 78% (and similar for s2 and s3).

This implies:

P(s1 is iid) = 1 - P(s1 is NOT iid) = 28%

Which is where I got the 28% from. Does that also seem okay? I will get to numerical results in the next post if this seems okay to you. Just for convenience, could you post your estimate of the probability Jesus existed assuming we only use 3 of the 15 points and the 78%?

[Bernard Muller]

to timhendrix,

Which is where I got the 28% from. Does that also seem okay? I will get to numerical results in the next post if this seems okay to you. Just for convenience, could you post your estimate of the probability Jesus existed assuming we only use 3 of the 15 points and the 78%?

That I can understand: P = 1 - (1-0.78)^3 = 0.989 = 98.9%

P(At least one of s1, s2, s3 is NOT IID) = 1 - P(s1 is iid AND s2 is iid AND s3 is IID)

As I see it:
P(At least one of s1, s2, s3 is NOT IID) = 7/8 = 0.875 = 87.5%
1 - P(s1 is iid AND s2 is iid AND s3 is IID) = 1 - 0.125 = 0.875 = 87.5%

Waiting for your next post.

Cordially, Bernard

[timhendrix]

to timhendrix,

Which is where I got the 28% from. Does that also seem okay? I will get to numerical results in the next post if this seems okay to you. Just for convenience, could you post your estimate of the probability Jesus existed assuming we only use 3 of the 15 points and the 78%?

That I can understand: P = 1 - (1-0.78)^3 = 0.989 = 98.9%

Waiting for your next post.

Cordially, Bernard

Just to make it very clear so we are not jumping between equations in a moment, the stuff I had in my previous post seemed okay?
Just to tie your number 98.9% into what I am doing. You start with is this (which I agree is 100% true):

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid)

Then you assume that (this is the controversial part)

P(s1 is iid AND s2 is iid AND s3 is IID) = P(s1 is iid)P(s2 is iid)P(s3 is IID) (The controversial step)

using you use that P(s1 is idd) = 28% (and so on for s2, s3) you therefore get that:

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid) = 1 - .28 x 0.28 x 0.28 = 1-0.28^3 = 97.8%

So asides the inequality sign (which I doubt will set us apart), I think it would be fair to say we agree on everything except the controversial step?

(updated: typo in the numbers).

[Bernard Muller]

to timhendrix,
I added that to my last post:

As I see it:
P(At least one of s1, s2, s3 is NOT IID) = 7/8 = 0.875 = 87.5%
1 - P(s1 is iid AND s2 is iid AND s3 is IID) = 1 - 0.125 = 0.875 = 87.5%

Then you wrote:

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid)

So I have P(Jesus existed) > 87.5%

For 4 events, instead of 3, then I would get: P(Jesus existed) > 15/16 = 0.9375 = 93.75%
Note 16 is 2^4
For 15 events, the we would get P(Jesus existed) > [(2^15) - 1]/(2^15) = 0.999969 = 99.9969%.

The problem is that way to calculate (if valid) does not take in account the probability of iid, such as 28% per point:
P = 1 - (1 - 0.72)^15 = 99.9999995%
OR
for iid = 22% per point:
P = 1 - (1 - 0.78)^15 = 99.999999986%

If I had iid = 80% per point, then:
P = 1 - (1 - 0.20)^15 = 96.48%
In that case, the result of your (controversial) equation for P(Jesus existed) > 99.9969% would not be compatible with the result from my calculation (96.48%).

But maybe you want to use the results of the two equations as inputs of a conditional probability equation?

Cordially, Bernard

[timhendrix]

Hi Bernard,

Just to be absolutely clear going forward, I think we agree this equation is always true:

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid)

So all our discussion can be limited to the term: P(s1 is iid AND s2 is iid AND s3 is iid)?

With regards to your post, I am not sure I understand what you are getting at here:

P(At least one of s1, s2, s3 is NOT IID) = 7/8 = 0.875 = 87.5%

(where does the 7 and 8 come from?).
Going forward in the discussion about P(s1 is iid AND s2 is iid AND s3 is iid), I suggest we distinguish between what follows from a logical perspective (i.e. what probability theory tells us this term can be given that we know P(s1 is IID) = 28%, P(s2 is IID) = 28% and P(s3 is IID) = 28%) and what "seems reasonable" or can otherwise be argued using a historical argument. I am only concerned about what follows logically at this point.

[Bernard Muller]

to timhendrix,
I based my understanding on your equation : P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid) = 87.5% because we have 8 combinations: 000, 001, 010, 011,100, 101, 110, 111 but there is only 1 combination out of 8 (111) where si, s2, s3 are all iid
That's why I concluded:

So I have P(Jesus existed) > 87.5%

because 1 - 1/8 = 7/8

Going forward in the discussion about P(s1 is iid AND s2 is iid AND s3 is iid), I suggest we distinguish between what follows from a logical perspective (i.e. what probability theory tells us this term can be given that we know P(s1 is IID) = 28%, P(s2 is IID) = 28% and P(s3 is IID) = 28%) and what "seems reasonable" or can otherwise be argued using a historical argument. I am only concerned about what follows logically at this point.

Because of your logical "AND", the overall result still will be for 1 - P(s1 is iid=28% AND s2 is iid=28% AND s3 is iid=28%) = 7/8 = 87.5%, assuming we have 3 probabilities with 28% iid.

Maybe you have in mind "AND" meaning addition OR multiplication?

Cordially, Bernard

[timhendrix]

to timhendrix,
I based my understanding on your equation : P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid) = 87.5% because we have 8 combinations: 000, 001, 010, 011,100, 101, 110, 111 but there is only 1 combination out of 8 (111) where si, s2, s3 are all iid
That's why I concluded:

So I have P(Jesus existed) > 87.5%

because 1 - 1/8 = 7/8

Oh, I see how that could make sense. Unfortunately, it is not true: there is no "rule" in probability theory that says that if we ask the probability A, B and C all happen that probability is 87.5%. Think about it, if that was true we could conclude:

P(I win the lottery tomorrow & there is life on mars & Magnus Carlsen will win the world championship in chess) = 87.5%.

Which is obviously not the case. Also think about what that case would say in case we only considered 2 statements or 1 statement:

P(s1 is iid AND s2 is iid) = 3/4 = 75%
P(s1 is iid) = 1/2(!?)

However, do I understand you correctly that you do not accept that we should use the equation:

P(Jesus existed) > 1 - P(s1 is iid AND s2 is iid AND s3 is iid) ?

If so, is that because you feel a part of my derivation was false?
(it is, of course, essential that we agree which equation we use and how it is derived before we can discuss if it is correct or not).

[Bernard Muller]

to timhendrix,

Oh, I see how that could make sense. Unfortunately, it is not true: there is no "rule" in probability theory that says that if we ask the probability A, B and C all happen that probability is 87.5%. Think about it, if that was true we could conclude:

P(I win the lottery tomorrow & there is life on mars & Magnus Carlsen will win the world championship in chess) = 87.5%

Actually, it should be P = 1 - (I win the lottery tomorrow & there is life on mars & Magnus Carlsen will win the world championship in chess) = 12.5%
Even so, that does not make sense. But it is your equation, not mine.
Personally, I would calculate your case with that equation: P = 0.0001 * 0.5 * 0.8 = 0.00004 = 0.0004%
assuming your chance to win the lottery tomorrow is 0.01%, life existing on Mars (as a microbial form) being 50%, and Magnus Carlsen winning the next chess championship (he won the last one) being 80%.
For 3 probabilities of 28% to be all true: P = 0.28^3 = 0.0018 = 0.18 %
For 15 probabilities of 28% to be all true: P = 0.28^15 = 0.0000000050976554 = 0.00000050976554%
For 15 probabilities of 72% to be all true: P = 0.72^15 = 0.007244150201409 = 0.72441502014095%

cordially, Bernard