Bayes and unconditional love.

So a friend of mine who is a heavy smoker told me that he is immune to lung cancer because it all depends on whether you believe in it or not. Being his personal data scientist, and from my unconditional love for my friend and data science I dove into reading more about the correlation between religion and human sicknesses. Having found several works on the Internet, I would like to expand more on theory of probability. Especially, the conditional probabilities.

Have you ever played a shell game?

So, the rules are thus: the host of the game shows that there is just one ball and three shells that he shuffles quickly and if you were inattentive, you’d probably come to guessing. So, in the first round you guess, but instead of opening the shell you just pointed at, the host suggests you that he opens one of the other two to show that the one he just opened does not have a ball and suggests you to either stay with the one you have just pointed to or switch to the other one he did not open.

A great majority of people, who basically are left to choose between two shells think that the chances to guess right are 50/50. And they prove wrong.

It turns out that the chances to guess right if you stay with your initial choice are a half of chances if you switched to the one neither you nor the thimble-rigger did not choose. Let’s see how the math works.

Let’s assign each shell a name: A, B, and C and say you pointed at A. Chances of a ball to be under one of those are 1/3. Let’s split and consider each case:

Without further developing, given that your chances to guess right are 1/3 only, in 2 out of 3 cases, you guessed wrong in the first place. And when the game owner opens you one of the left two, serves you are great service — you still are choosing between your 1/3 and 2/3. So, now your call. Would you stay with your original choice or switch?

Mathematically it is expressed with Bayes theorem:

P(A|B) = P(B|A)*P(A)/P(B)

In English it reads as follows: probability for an event A, given that B has happened equals the probability of event B, given that A has happened, multiplied by the probability of event A alone and divided by probability of event B alone.

Here probability of event A happening is called prior, probability P(B|A) is called likelihood, probability of B (P(B)) is normalizer or marginal likelihood. The resulting P(A|B) is a posterior.

So, stop smoking my dear friends, since my unconditional love to you ends with your life.

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