De Méré Paradox

The De Méré Paradox is a mathematical paradox of probability theory from the 17th century, which was named after Chevalier de Méré .

History of the de Méré paradox

When the then well-known French gambler Chevalier de Méré met Blaise Pascal , a scientist and mathematician who was highly regarded at the time , he asked him a question about gambling. When Pascal presented his answer to him, he was not particularly surprised because he already knew the answer. Pascal solved the problem, but not the apparent contradiction.

The paradox

Once you throw a six-sided dice (which, if it is perfect, a Laplace dice is), is the probability of rolling a 6, a sixth.

If you throw two such Laplace dice once, the probability of rolling a double six is six times lower than the previously mentioned probability, namely 1/36.

If you roll a Laplace dice 4 times, the probability of rolling at least a 6 is just over 50%.

If you roll the two Laplace dice 24 times, the probability of rolling a double six at least once is just under 50%.

The paradox is that the probability of success per throw in the last experiment is exactly one sixth of the probability of success per throw in the penultimate experiment, but the number of throws is six times as large. On a cursory level, one could therefore assume that this compensates for each other and that the success probabilities in the last two experiments are the same.

On closer inspection, however, this is not the case.

Explanation of the paradox

When trying with the 4 throws is

{\ displaystyle P ({\ text {at least one six}}) = 1-P ({\ text {no six}}) = 1- \ left ({\ frac {5} {6}} \ right) ^ {4} \ approx 0 {,} 5177 \ approx 52 \, \%}

When trying with the 24 throws is

{\ displaystyle P ({\ text {at least one double six}}) = 1-P ({\ text {no double six}}) = 1- \ left ({\ frac {35} {36}} \ right) ^ {24} \ approx 0 {,} 4914 \ approx 49 \, \%}

This did not surprise or satisfy de Méré because he already knew this result.

He wanted to have resolved the contradiction why the results did not behave proportionally as . ${\ displaystyle 4: 6 = 24: 36}$

In the book “Doctrine of Chances” published in 1718, Abraham de Moivre pointed out that the “rule of proportionality of critical values is not far from the truth”.

“Critical value” means the minimum number of throws that is necessary for the attempt's chance of success to be over 50%. ${\ displaystyle n}$

The critical value is the smallest natural number for which applies , synonymous with ${\ displaystyle n}$ ${\ displaystyle 1- (1-p) ^ {n}> {\ frac {1} {2}}}$

${\ displaystyle n> {\ frac {\ ln {\ frac {1} {2}}} {\ ln (1-p)}} = {\ frac {\ ln {\ frac {1} {2}}} {-p - {\ frac {1} {2}} p ^ {2} - {\ frac {1} {3}} p ^ {3} - \ ldots}} = {\ frac {\ ln 2} { p + {\ frac {1} {2}} p ^ {2} + {\ frac {1} {3}} p ^ {3} + \ ldots}}}$ .

The logarithmic power series expansion was used here.

Using Landau's symbols , the last term can be written as . ${\ displaystyle {\ frac {\ ln 2} {p + {\ frac {1} {2}} p ^ {2} + {\ mathcal {O}} (p ^ {3})}}}$

It can be seen that there is no proportional relationship, but that a quadratic term is relevant, among other things. The approximation improves the more terms are taken into account. The proportionality can be used as a first approximation, but does not give exact results.

literature

Andreas Büchter, Hans-Wolfgang Henn: Elementary Stochastics: An introduction to the mathematics of data and chance. Springer, Berlin Heidelberg 2005, ISBN 3-540-22250-2 , pp. 221-223.
Wolfgang Göbels: The problem of the Chevalier de Méré. German Association for the Promotion of Mathematics and Science Education 66/1 (January 15, 2013) pp. 12–13, ISSN 0025-5866, © Verlag Klaus Seeberger, Neuss.