## Innumeracy

In one study, Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she’s in a low-risk group: 40 to 50 years old, with no symptoms or family history of breast cancer.  To make the question specific, the doctors were told to assume the following statistics — couched in terms of percentages and probabilities — about the prevalence of breast cancer among women in this cohort, and also about the mammogram’s sensitivity and rate of false positives:

The probability that one of these women has breast cancer is 0.8 percent.  If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram.  If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram.  Imagine a woman who has a positive mammogram.  What is the probability that she actually has breast cancer?

Gigerenzer describes the reaction of the first doctor he tested, a department chief at a university teaching hospital with more than 30 years of professional experience:

“[He] was visibly nervous while trying to figure out what he would tell the woman.  After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent.  Nervously, he added, ‘Oh, what nonsense.  I can’t do this.  You should test my daughter; she is studying medicine.’  He knew that his estimate was wrong, but he did not know how to reason better.  Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities.”

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## Stare at the dot

This is a lovely and somewhat strange illusion. Stare at the red dot in the left-hand image for about 20 seconds, and then look at the centre of the right-hand image. All being well, lots of strange stuff will happen in your brain…..

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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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## An amazing optical illusion

Here is an amazing optical illusion:

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## Ames’ Window: A magnificent visual illusion

This perspective window by Adelbert Ames upsets size-shape scaling to produce remarkable effects.”

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## A persistent optical Illusion

The “gateway arch”  is as wide as it is high. Simply use a ruler to confirm its dimensions!

Humans seem to subconsciously heighten shapes that soar skyward and thin rapidly as they rise. Amazing ehhh?

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