## 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):

.

1) HHHHTTT

2)THHTHTT

3)HHHHHHH

4)TTTTTTH

.

.

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

.

.

III)

You are a member of a jury. A taxi driver is accused of having run down a pedestrian on a stormy night and having fled the scene of the accident. The prosecutor, in asking for a conviction, bases his whole case on a single witness, a lady who saw the accident from her window a little way away. The lady testifies that she saw the pedestrian struck by a blue taxi and then saw that taxi drive away from the scene of the accident. The accused works for a taxi company whose taxis are all blue. During the trial the following emerges:

1. There are only two taxi companies in this town. The whole fleet of one company is green; the other has only blue cabs. Eighty-five percent of all taxis on the road that night were green, and only 15 percent were blue.

2. The single witness has undergone a number of vision tests in conditions similar to those of the night of the accident. She has been shown to be able to distinguish a blue taxi from a green on 80 percent of the time.

On the basis of the sworn testimony of the witness, and the data offered in 1 and 2, what is the probability that the taxi was really blue?

What say you jury?

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Source: The interesting book of Massimo Piattelli-Palmarini, Inevitable Illusions: How Mistakes of Reason Rule Our Mind

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## 6 thoughts on “Probability Games (Do you suffer from probability illusions?)”

1. Ⅰ) p₁ = p₂ = p₃ = p₄ = 2⁻⁷.

Ⅱ) p₁ = 2·3⁻⁵ > p₂ = 4·3⁻⁶ > p₃ = 2·3⁻⁶.

Ⅲ) p = 0.15 / (0.15·0.8 + 0.85·0.2) = 0.15 / (0.12 + 0.17) = ¹⁵/₂₉.

• epanechnikov says:

I) ✓

Experiments have shown that most subjects think the second sequence (which appears to be the most typical) is the most probable.

II) ✓

Note that the ordering can be done without calculations. Sequence 1 is a subsequence of sequence 2 and hence it is more likely. All but one of the elements of sequence 3 are identical to the ones of sequence 2. And since G is more likely to appear than R the sequence 2 is more plausible.

Due to their belief in the law of small numbers people tend to think that the sequence 2 is more plausible.

III) x

P(B|W+) = P(BΛW+)/P(W+) = P(BΛW+)/(P(W+ΛΒ)+ P(W+ΛG))

• You are right. Originally I had multiplied the nominator by 80%, but then I removed the factor, although it’s necessary because we are interested not just in blue taxis (15%), but in blue taxis that the witness saw as blue (80% of 15%).

• epanechnikov says:

It is interesting that the posterior probability for blue is considerably lower than the one for green (41% vs 59%). Most people get confused and fail to consider the base rates.

• I should read Daniel Kahneman’s books…

• epanechnikov says:

Sure. Gerd Gigerenzer has a few good books too. 😉