Vsub Segment 2

Calculating

1. If the knight had captured a Gnome instead of a Troll, what would his chances be of crossing safely? Pr( Bridge is safe | Knight captured gnome) = = 0.682

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?

Pr(Ball is blue) = P(Box=A, Ball=Blue) + P(Box=B, Ball=Blue) If it is blue, what is the probability that it came from box B?

Pr(Box=B|Ball=Blue) = P(Ball=Blue|Box=B) P(Box=B) / P(Ball=Blue) 1. Do you think that the human brain's intuitive "inference engine" obeys the commutativity and associativity of evidence?

I guess not.

For example, are we more likely to be swayed by recent, rather than older, evidence?

Yes, I think our brain weighs recent events more heavily than older evidence.

How can evolution get this wrong if the mathematical formulation is correct?

a) For simpler problems, it's likely that the mathematical formulation is valid for a one-shot game, but the brain thinks about it as a game that's played multiple times. b) The brain is not very good with handling too many variables and knobs. Intuition works well on simpler problems with fewer variables.

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?

a) Pick bridge 1 and 2 with equal probability.

b) Pick a creature from the bridge (based on the probability distribution)

c) Consider this sample as valid only if the creature picked from (b) is a troll (otherwise discard). Check if the knight crosses safely on this sample.

d) Count the number of (valid) samples on which the knight crosses safely when the count of valid samples reaches 100,000.

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?

You might have to make assumptions, such as there is some background information that's common to all observers.