Segment 2. Bayes
1. If the knight had captured a Gnome instead of a Troll, what would his chances be of crossing safely?
Using Bayes Law
we have that the probablity we are on bridge type is
and the probability we are on bridge type is
and the probability we are on bridge type (the probablity he can cross safely) is
Cross safely, good sir!
2. Suppose that we have two identical boxes, A and B. A contains 5 red balls and 3 blue balls. B contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box. What is the probability that it is blue? If it is blue, what is the probability that it came from box B?
L - probability the ball drawn is blue.
By the law of de-anding, the probability the ball drawn is blue is
By Bayes law, the probability we are in box B given the ball is blue is
To Think About
1. Do you think that the human brain's intuitive "inference engine" obeys the commutativity and associativity of evidence? For example, are we more likely to be swayed by recent, rather than older, evidence? How can evolution get this wrong if the mathematical formulation is correct?
No, one example of being swayed by more recent evidence is in Texas Hold Em poker, in which the final community cards are laid out one at a time. Although you should make your decision mostly on your first two cards since those are the one advantage you have over others, many people feel a rush when their odds of having a good hand dramatically increase with the next 5 cards.
An example of being swayed by older evidence rather than more recent, conclusive evidence is the pervasive view propagated during the 1990's that a low-fat diet was a straightforward route to preventing heart disease, some cancers, and the epidemic of obesity. Most studies and metastudies now show that refined carbohydrates are the far worse evil, and some even show that a low-fat diet has no impact whatsoever. Yet almost everyone I talk to has some confusion about which is worse - probably because many high carb processed foods like white bread still have labels that proclaim "reduced fat!"
2. How would you simulate the Knight/Troll/Gnome problem on a computer, so that you could run it 100,000 times and see if the Knights probability of crossing safely converges to 1/3?
3. Since different observers have different background information, isn't Bayesian inference useless for making social decisions (like what to do about climate change, for example)? How can there ever be any consensus on probabilities that are fundamentally subjective?
But isn't there a way for us to measure how much background information we have? We need to remember that all issues are contentious mostly for emotional reasons, which directly conflict with the rational part of us. Even though economists admit that they assume rational behavior, they still have children even though the majority of studies have shown that regardless of socioeconomic status, couples who have children are less happy than those who don't.