Tag Archives: Probability

Risk: The Neural Basis of Decision Making

“Lecture presented by Professor John O’Doherty for the Darwin College Lecture Series 2010.

A deeper understanding of how the brain makes decisions will not only inspire new theories of decision making, it will also contribute to the development of genuine artificial intelligence, and it will enable us to understand why some humans are better than others at making decisions, why humans with certain psychiatric and neurological disorders are less capable of doing so, and why under some circumstances humans systematically fail to make rational decisions. Most decisions made in everyday life are taken for the purposes of increasing our well-being, whether it is deciding what item to choose off a restaurant menu, or deliberating over what career path to follow. Prominent amongst these is the value or utility of each decision option, which indicates how advantageous a particular option is likely to be for our future well-being. Another relevant signal present in the brain is the riskiness attached to a particular decision option, which can influence the decision making mechanism according to ones own individual preferences (whether one is risk-seeking or risk-averse).”

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Probability Games (Do you suffer from probability illusions?)

I) I have just tossed a fair coin 7 times. Guess which of the four sequences below is genuine (order them according to their plausibility):

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1) HHHHTTT

2)THHTHTT

3)HHHHHHH

4)TTTTTTH

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II) A die has been painted in such way so that it has four green faces and two red. I have thrown the die repeatedly into a table and produced one of the four sequences. Which one is the genuine one (please order them according to their plausibility)?

1.RGRRR

2.GRGRRR

3.GRRRRR

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The Monty Hall problem

Consider the following problem
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

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(Formulation by Marilyn vos Savant at Whitaker, Craig F. (1990).)
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So what is the answer? Let’s see a simple solution. The following table shows the three possible arrangements of two goats and one car behind three closed doors and the result of switching or staying after initially picking Door 1

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Door 1 Door 2 Door 3 Outcome if switching Outcome if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat

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Which means that in 2 out of 3 cases we would win the car by switching doors. Hence it would make perfect sense to always switch. To understand why this is happening do take into consideration that the event “staying” is composed by only one element while its complement by two (note that all elements are equiprobable).

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Frequency of winning when staying/switching doors in a sequence of "Monty-Hall" games

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The principles of Frequency Probability (Von Mises)

1) Unlimited Repetition : The theory of probability applies only to a practically unlimited sequence of uniform observations. Hence, the probability applies only to problems in which either the same event repeats itself, or a great number of elements are involved simultaneously. The probability, in a frequentist sense, is not synonymous to trustworthiness or to the degree of belief.

2) The Collective: A probability is attached to a class or a collective but nothing can be said about the probability of a single member of the collective.

3) Relative Frequency: The relative frequencies of certain attributes in a collective become more and more stable as the number of elements is increased and will converge to a fixed limit. This limit is “the probability of the attribute considered within the given collective”.

4) The Principle of Randomness: The limiting values of the relative frequencies must remain the same in all subsets which may be drawn from the original sequence in an arbitrary way. It is however crucial that the probability of the attribute considered in a collective is independent of all possible place selections. Note that if a sequence satisfies conditions 1-3 but not condition 4 then the limiting value will be called “the chance” of the occurence of the attribute rather than its “probability”.

The aforementioned principles of frequency probability are presented in a classic work of one of the most influential figures of the frequentist school of statistical though, Richard Von Mises (see Richard Von Mises (1957), “Probability, Statistics and Truth”, Dover Publications).

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update: Michael P. O’Brien provides some further explanation of my summary  here.

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