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Take as an example the Eurojackpot, where 5 randoms (ranging 1-50) are drawn from a pot with yellow balls, and 2 bonus balls (ranging 1-10).

Video of the drawing

  1. How do lotteries ensure that the drawings are sufficiently random? Is there a way to test how random the drawings are?

  2. What are some approaches to find possible vulnerabilities? Are there any statistical methods known? How would one go to exploit this info?

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It really depends on the meaning you are giving to the word "random". The wikipedia definition of randomness is

Randomness is the lack of pattern or predictability in events. A random sequence
of events, symbols or steps has no order and does not follow an intelligible 
pattern or combination.  

The problem with this definition is that it is too broad for a single statistical test of randomness.

To overcome this difficulty we do not test if a sequence is random, but if it follows a particular probability distribution.

Consider the example of a casino roulette where we are interest in deciding if the results are trully random, in the sense that getting a odd number is equally probable of getting a even number.

In this setting we can define the variable $X = 1$ if result is odd and $X = 0$ if the result is even. And besed on a set of observations $\{X_1,\ldots,X_n\}$ use a statistical test to check if we can reject the hypothesis that the probability of $X = 1$ is equal to that of $X = 0$. In the case where this hypothesis is rejected we can confidently affirm that the well is not fair.

For more general situations you might want to check some goodness of fit tests.

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  • $\begingroup$ Lotteries can be exploited even when they adhere to a "particular distribution," if there is any serial dependence over time among their results. Thus the relevant sense of "random" ought to be wider than the limited one you propose, unless you can argue successfully that all draws in all lotteries are independent. $\endgroup$ – whuber Feb 23 '17 at 17:10
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For the German lottery, it is said that the lottery balls with one numeral are labeled with 15 levels of paint; while balls with two numerals are labeled with 12 levels in order to make sure the balls have nearly equal weight.

Also in contrast to the early years, the numbers are not drawn by hand anymore in most lotteries.

The randomness has been examined for example by Harry Joe in "Tests of uniformity for sets of lotto numbers", Statistics & Probability Letters 16, 1993, 181--188 and by H. F. Coronel-Brizio et al. in "Statistical auditing and randomness test of lotto k / N-type games", Physica A 387 (2008) 6385–6390.

They did find deviations between theoretical and observed frequencies in some lotteries. They did not develop strategies to exploit that, though.

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