Bayesian Probability is like a reaction to the Mathematical Probability: what about our intuition whilst experiencing the event?
Most of the people would not be happy if told that, despite having 50% of mathematical chance of getting the car either way (switching or sticking), the best approach is swapping on the second question, since, in practice, in that particular show, people do get the car more times when they switch than when they stick.
Once more, that is what distinguishes Mathematics from Decision Theory/Management.
Human beings may learn Mathematics well, but they frequently go with their intuition: I know that I could cover more options with a trifecta, but my friend, he did like this, and won. I trust his wisdom.
There is an expectancy of winning if everyone who used the red ball won that day even if Mathematics says that the chances are all the same, regardless of the ball chosen.
In this case, the red ball moving forms evidence in the direction of the player winning: if we see a red ball moving, we believe the probability that the person win is now superior to the usual probability, of those who pick a ball of any other colour, even though the mathematical probability says that all colours have the same chance.
If the probability of this event we observed (red ball moving, winning) is greater than the probability of the connected event of maximum scope (a ball moving, winning), then we say that E (red ball moving, winning) confirms H (a ball moving, winning).
We now see a decision being given, something belonging not only to Decision Theory theories, but also to Philosophy of Science: I see Hamish from where I am, inside of the same room, and that might indicate that Hamish ‘is with me’. The probability that ‘Hamish be with me’ if I know that Hamish is now in the same room as me might be increased, but it may also be that I am in this room to be bashed, and everyone who is in the room that is not me is ‘against me’ instead. All depends on context, and we usually need more than three facts to work with probability inside of the World of Humans, since there everything is very different from what is inside of the World of Mathematics. Notice however that, if we reason like that, probability becomes personal, tailored to the logical paradigms of the decision maker: since I believe that the simple fact that Hamish is in the same room as me means that he is with me, if I am the decision maker, I will say that the probability that he is with me increases if he is in the same room as me. If the other person, who has this alternative scenery inside of their Inner Reality, is assessing things instead of me, they will say that Hamish is definitely not with me if he is in the same room instead. Bayesian probability gives us the freedom to reason in a non-mathematical way, but we then have nothing to work with apart from our own reasoning or our Private Logic.
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