June 30, 2006

Random draws: a statistician’s further insight

Editor’s note: Our article, “Championship draws: a statistical analysis,” considered the basic odds of occurrences in random draws for orders-of-play at pipe band competitions. It has created much constructive dialogue around the world, and we have received many excellent responses. All pipe band associations act completely openly, and draws are totally random. We have no reason to doubt that. Anything can happen when it comes to random draws from contest to contest. But each draw is carefully scrutinized by those competitors it impacts. While some agonize and invent conspiracy theories, other bands, as we heard from one reader, “just get on with it.”

We heard from Tim Murphy, a well known snare drummer who also happens to teach statistics at Brock University in Ontario. He provided further thoughts and analysis on draws and probabilities from contest to contest. We hope that readers would find them interesting.

I read your article regarding the statistical analysis of the draws for the Scottish and British Championships. The probabilities you cited have a few assumptions that are not clearly outlined and would change the results.

The odds of 1:156 are actually the probability that ScottishPower and Clan Gregor would draw first and last respectively in a field of 13 bands, not that this would happen twice in a row. In other words, the probability that the two bands who played first and last bands at the previous contest are first and last at the next contest. This may seem like just semantics but in math circles it is an important distinction. Here is simple example of what I mean. The probability of flipping a coin and getting two heads in a row is (.5)(.5) or .25 or 1:4. However, the probability of the second flip being the same as the first is simply .5 or 1:2.

If you approach this from an a priori point of view (i.e., before the draw for either contest), the probability of those two bands playing first and last respectively becomes 1:20,592 (1:132 for a field of 12 and 1:156 for a field of 13). However, the probability of any two bands being first and last on respectively in two fields of 13 is 1:132. I did two fields of 13, otherwise the math gets messy when one band is not at one of the contests.

As for Field Marshal Montgomery playing directly before House of Edgar-Shotts & Dykehead, you listed 1:132, however, there are 12 ways for that to happen, namely:

  • FMM 1st, and Shotts 2nd, with the other bands filling out the field
  • one band, then FMM 2nd and Shotts 3rd, then the rest, etc.
  • up to, the other 11 bands in any order, then FMM 12th and Shotts 13th.

    So, I think the actual odds of two bands playing in the same order two weeks in a row (from two fields of 13) are 1:13. The probability that they drew not only one after the other, but in exactly the same position (i.e., 5th and 6th both times respectively) would be 1:132.

    This is similar in general to a famous math problem. If you have 23 people in a room what is the probability that two will have the same birthday?

    Amazingly, it turns out to be .5. In other words, it happens about 50 per cent of the time.

    However, the probability of selecting one person and then having another with the same birthday is about .05. In other words, it only happens about 5 per cent of the time. It is all based on conditional probabilities (i.e., do you know the first outcome or not).

    You also asked about the incidence of set/medley #1 being drawn more often than would be expected. If anyone has access to this data, it could easily be verified (and I would be happy to do it). However, assuming this information is not readily available, here are some estimates of the outcomes that would be required to have statistical evidence that these “random” draws are not quite so random.

    If you have 20 bands in a contest it would take 15 out of 20 to draw set/medley #1 to be considered non-random (significant) at the accepted standard of a probability less than .05 (i.e., 95 per cent) confidence level.

  • If you have 30 bands in a contest it would take 21 out of 30.
  • If you have 40 bands it would take 27 out of 40.
  • Now, if you pool across several grades and/or contests and have, say, 100 data points, then it only takes 60 out of 100.
  • Collect for an entire season and have 500 data points and you are down to needing 272 out of 500.

    These are just some examples to show you the types of ratios of how often set/medley #1 would have to be drawn to be significant. If this data exists, I can easily do a proper analysis.

    If you are ever suspicious that a band seems always to draw their favoured set/medley a bit too often, then determining the likelihood of that happening simply by good fortune is fairly straightforward.

    If the draws are truly random then at each contest the Pipe-Major has a chance of .5 of drawing the bands #1 set/medley To calculate the probability of several favourable draws in a row is simply .5n where n is the number of times it has happened

    Therefore, the probability of this happening:

  • Twice in a row is .52 = .25 or 1:4
  • Three times in a row is .53 = .125 or 1:8
  • Four times in a row is .54 = .0625 or 1:16
  • Five times in a row is .55 = .03125 or 1:32
  • Six times in a row is .56 = .0156 or 1:64
  • etc.

    As you can see, once this has happened about four or five in a row you may start to wonder.

    Actually, it was a question similar to this that led to the creation of statistics.

    Fisher (a famous statistician, not the drummer) had a lady friend who insisted that when she was served tea that the milk be put in first, then the tea. She claimed to be able to taste the difference (she did not like milk added to tea). Fisher doubted there was any noticeable difference and decided to test her.

    He realized that correctly identifying how her tea had been prepared once would not be that convincing (50 per cent chance of being right), so he began to play with numbers and decided that if she could do this eith times in row he would be convinced. The story is that she correctly identified all eight and he was impressed.

    He then began to contemplate the actual probabilities and decided that about four or five times would have been impressive enough, but since .0625 and .03125 were awkward numbers to deal with, it eventually became standardized at .05 as the point at which we have faith in an effect. In other words abandon the hypothesis that this is a random event. As can be seen above, sometimes seemingly non-random events (such as 14 out of 20 bands drawing set/medley #1) are still not rare enough to warrant being called statistically significant.

    I hope readers find this at least marginally interesting.


    Tim Murphy has taught statistics and research design at Brock University in Ontario for the last 11 years. He has a B.Math from the University of Waterloo, a BA and MA from Brock University, and will complete his PhD at the University of Waterloo this year. A side drummer, he played with the 78th Fraser Highlanders from 1983 to 1992, and before that with the Niagara & District Pipe Band.


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