A tombola with a bias

The other day I attended a street festival in a nearby street here in Cologne. Like any other street festival it offered a range of activities: home made food and cakes; flower headband workshop; and, of course, a tombola (also known as raffle in some places).

Me, my partner and her brother purchased 5 tickets at the tombola the cost of 1€ each. There were 25 possible awards and 2,000 available (and sold) tickets. From the outset, I knew our chances were limited. In fact before the draw took place, I calculated that we had around 0.25% chance of winning any specific award and a 6% chance of winning any award. Given that the most expensive price of the Tombola was a 500€ bike it was rather clear that my bet would always have a negative expected value. But, in a local street festival you play to play, not to win.

Wheels of fortune

To determine the winner of each tombola price the tombola masters had built a brilliant lucky wheel device. The way the device worked was that the tombola master would spin four different lucky wheels. The first three wheels all looked the same with the numbers 0 to 9 written on the and separated in the same way a dartboard is. The wheels were ordered from the left to the right and once all three wheels stopped spinning a three digit number would form. The fourth wheel was split in two where one side was green and the other red. this would then determine the colour of the winning ticket.**

Me, my partner and my brother had purchased green tickets in the number row 455 – 459. While we were not expecting to win, we did expect to come close at least few times. At least we were fairly confident that at least one time 4 would come up on the first wheel and in fact, we had good hopes that at least once in the draw 4 would come up on the first wheel as well as our colour, green, would also come up. Or that the number 4 would come up on the first wheel as well as 5 on the second wheel.

1

Our winning chances were not great, but the chance of being in the right hundred were high.

Now, this is important because in the Tombola, how the wheels worked was that generally the first wheel, indicating which hundred the winning ticked belonged to, stopped first.  At that point, the probability of us winning had either gone down to zero (in case 4 did not come up) or massively improved. In fact, in any roll, if 4 came up first then our chance in that role would instantly rise from 0.25% to 5%. Now, in the less likely event of the second number also being our number (5) then our chances would explode to 25%. And I must admit that the greater the chance to win the more fun I have.

Numbers from 400 – 500 never came up

While it is no surprise we did not win it was a surprise that no number in the 400 to 500 range ever came up. The chance of this happening is only about 7%. Which is unlikely, but no doubt reasonable. In fact, in a random tombola with the same design as this one, we would expect this to happen around once per 14 tombola.  But there was more to it. when I analysed the final winning numbers a bit further I noticed that there was something fishy going on with the first two wheels.

The first wheel, it turned out, seems to have a massive bias towards the numbers 0, 1 and 2. These numbers came up 22 out of the 25 times. This bias lead to that 3, 4 and 5 did not come up once. The odds of non of the numbers 3,4 or 5 coming up ever in 25 spins is extremely low. In fact, we would only expect that to happen in one per 7,457 tombolas. Of course this could be a result of chance, but I highly doubt it (after all the wheel had a tendency to roll back from 5 to 1 or even zero).

IMG_5614

The winning numbers of the tombola.

The second wheel had a similar bias, although our number (5) came up once, the wheel seemed to dislike the numbers 6,7 and 8 as they never came up once during the tombola. Again, the chances of this happening is the same as in the first wheel. However, if you were to ask the question: “what are the chances that 3,4 and 5 never come up on the first wheel and 6,7 and 8 never on the second wheel, during 25 spins?”, the answer would be, 0.00002%. If a tombola, with a fair tombola wheel were to spin at 55 million tombola events, we would expect to see that pattern only once!

So, I must admit that by the time the tombola was half way through and only the low numbers had come up on the wheel, I lost a bit of interest in the tombola. Now, I do not think for a minute that the brilliant organisers intended to rig the tombola nor that they had the slightest clue of the bias in the wheels prior to spinning them.  In fact, I don’t even blame them for missing out on the bias in their cool design. After all I would rather play biased-wheels tombola any time over a random number generator from Excel tombola.

Thanks for reading,
Eiki

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*For example, if first wheel ended on 2, second wheel on 0 and third wheel on 7, the winning number would be 207. Then the fourth wheel would stop on either red or green, and if it stopped on say red, the winner would be 207 RED.

 

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