Lotto silliness

As my good friend and colleague Thomas Lumley points out we have plenty of Lotto-based silliness to tide us over until the next stupid health related press release from a conference with no quality checks. Case-in-point is the article Powerball could be in the stars in Thursday’s NZ Herald (29 March, 2012, A5): (also nominated by Sammie Jia for Stat of the Week).

The article reports the frequency of zodiac signs from a survey of 104 first division Lotto winners, and gleefully touts Taurus as the luckiest star sign with 13% of the total. The article gives us a summary table:

Of course, all keen Statschat readers will note that this table does not add up to 100%, nor does it show all twelve zodiac signs, which is not very helpful. Buried in the text is the additional information that Aries and Cancer combined make up 4% of the total.

If we spread the remaining probability over Gemini, Sagittarius and Scorpio, and make the not entirely justified assumption that the distribution of zodiac signs is uniform (which is exactly what the NZ Herald has done), then we can perform a simple chi-squared test of uniformity. This yields a P-value of 0.22, which for most frequentists isn’t exactly compelling evidence.

Being a Bayesian, I prefer to assume multinomial sampling with the prior on the probability of success being uniform. The figure below shows posterior credible intervals (based on 10,000 samples) for the true probability of success. The red dots are the observed values. The dashed line is the equal probability line (0.083 = 1/12).

All of the intervals overlap confirming our statistical intuition that all we are really observing is sampling variation. Yes, Ares and Cancer do fall below the line, but they are not significantly different from the other signs. You can, of course, not believe me – in which case Thomas has some tickets from last week’s draw going very cheap and your chance of winning is almost the same.