Betting on horses and the resurrection of Jesus (II)

Versione italiana

Thomas Bayes (1701–1761)

In the last article I used a horse racing / mathematical problem to introduce Bayes' theorem in its odds-based form, ie the one using the ratio of the probability p that an event 'A' occurrs and of the probability 1-p that it does not occur: O (A) = p / (1-p).

At the end of that same article, I also promised a connection between this formulation of Bayes' theorem and the resurrection of Jesus. Obviously the key to this promise is found in the article immediately before, "On the validity of the testimony of the apostles about the resurrection of Jesus", in which I explained why the testimony of the apostles of Jesus about his resurrection is not an evidence strong enough to accept as true the hypothesis that a human being has risen from the dead. Now, with the help of Bayes' theorem, we can calculate how unlikely this hypothesis is.

Bayes' formula

First a small summary. Let O (H) be the odds of the hypothesis 'H' regardless of the occurrence of event 'E', P (E | H) the probability of the event 'E' when the hypothesis 'H' is true, P (E | ¬ H) the probability of the event 'E' when the hypothesis H' is false, and O (H | E) the odds of the hypothesis 'H' when event 'E' is taken into account; then Bayes' theorem is expressed in the following form:

O (H | E) = O (H) * P (E | H) / P (E | ¬ H),

that is the odds of hypothesis 'H' when observing event 'E' are equal to the odds of hypothesis 'H' regardless of the observation of 'E' multiplied by the likelihood ratio, the ratio between the probability to observe the event 'E' when the hypothesis 'H' is true and the probability of the event 'E' when the hypothesis 'H' is false. Through this ratio it is possible to update the odds of a hypothesis following the observation of an event.

The subject of the inquire

Our hypothesis 'H' is "Jesus was truly risen", meaning that Jesus was a human being and that his body returned to life. We have no direct observation of this resurrection, but we observed the event 'E' "the apostles of Jesus were witnesses of the resurrection".

The question we ask ourselves is what is the value of O (H | E), what are the odds that the hypothesis "Jesus is truly risen" is true, taking into account the event 'E' "the apostles of Jesus were witnesses of the resurrection". Applying Bayes' theorem, we find that this quantity depends on three factors:

1. the odds that the hypothesis 'H' is true, regardless of the testimony of the apostles, O (H);
2. the probability that the apostles would have been witnesses of the resurrection if Jesus was really risen, P (E | H);
3. the probability that the apostles would have been witnesses of the resurrection if Jesus was not really risen, P (E | ¬ H).

How many people have lived on Earth?

What is the probability that Jesus rose from the dead, regardless of the testimony of the apostles? Given that this testimony is the only evidence of such an event, it is not unreasonable to think that the resurrection of Jesus has in general the same probability of occurrence of the resurrection of every other human being.

Betting on horses and the resurrection of Jesus (I)

Versione in italiano

A friend of yours, a real fanatic of horse races, has convinced you to follow him at the racecourse on a racing day. While you are waiting for the beginning of the races, you decide that, after all, a small betting would be fine. So you read the program of the first race, and the name of a horse comes to your attention: Soldatino!

You ask your friend if Soldatino is a good horse, but the answer is negative: in 105 races in which it participated, it "showed" (i.e., it arrived within the first three places) in just 7, that is, for each time it showed it did not for 14 times! Well, not exactly the favourite of the race...

Seeing your disappointment, your friend tries to get you a bit on the moral: "During the night it rained a lot, so the ground is very heavy." Noting your quizzical expression, your friend tells you that in 70% of the races in which Soldatino showed, the ground was heavy, and in those races in which it came in fourth or worse, the ground was heavy in 10% of cases.