What are my chances of winning?

Photo by Annie Spratt on Unsplash

One of the key skills for a betting person is the ability to assess the worth of a bet.
Assessing probabilities is usually one of those situations where we default to gut feeling and that gut feeling is usually wrong.

Simple example

Let us take a simple example.
You are offered a bet of calling the flip of a coin. If your guess is correct you win £10.
To enter into this bet you pay £5. Would you take the bet?
Why?/why not?

To get your answer you need to calculate the "expected value" (EV) of your bet.
The formula is: EV = (probability of winning * amount won) - (probability of losing * amount lost).
In this case EV = (50% * £10) - (50% * £5) = £2.5.
When EV is positive you are expected to come out profitable over the long run.
So yes, you should take the coin-flipping bet ... Even if you had to pay £8 (I know your gut feeling is protesting now).

BOTB example

Okay, so we are not wired to assess bets with our gut. By now you are wondering what the EV of your BOTB tickets is.
Or, more precisely, if the EV is positive.
So lets figure out when we cross into favourable bets territory by solving for an EV of 0:

According to BOTB the worth of one Lamborghini Urus S ticket is £5.79 and the car itself it worth £190,000.
For an EV of 0 we get:

  • 0 = (probability of winning * £190000) - ((1 - probability of winning) * £5.79)
  • £5.79 = probability of winning * £190005.79
  • probability of winning = £5.79 / £190005.79
  • probability of winning ≈ 0.0000304206
  • probability of winning ≈ 1/30421

So if your probability of winning is better than 1/30421 you are entering into a bet that will be profitable over the (very) long term.

So let us say your bet should be 1 pixel out of 30421 pixels. You get the picture now, right?
Basically a circle with an area of 30421 pixels has a radius of 98 pixels. In other words if you get closer than 98 pixels you are entering into a favourable bet.

Zones and credit

Now lets factor in zones and credit won.
Since we are entering into multiple bets over the long term I will take the liberty to simply deduct credit won from the ticket price.

  • Zone 1 = 100% credit (Area: 7854px).
  • Zone 2 = 50% credit (Area: 31416px including zone 1).
  • Zone 3 = 15% credit (Area: 402639px including zones 1 and 2).

Obviously getting zone 1 is favourable because you pay nothing for a 1/7854 probability.
Zone 2 is likewise favourable because you now pay only half price for a probability that is roughly equal to the break-even EV of a normal ticket price like we calculated above.

The real question then becomes Zone 3:
The area of Zone 3 = 402639px - 31416px - 7854px = 363369px.

  • EV(Zone 3) = (1/363369 * £190000) - (363368/363369 * (0.85 * £5.79)).
  • EV(Zone 3) = 0,52 - 4.92.
  • EV(Zone 3) = -4.40.

if you consistently get Zone 3 you are not entering into favourable bets - or in other words you are losing money over the long run.

When will I win?

I think many serious BOTB players that use community averages etc. manage to keep inside Zone 1 and 2 - with the occasional Zone 3 spells.

For the sake of this example let us look at our chances if we manage to keep inside zones 1 and 2.
You are playing an area of 31416 pixels. Again, let us assume you play 20 tickets every week. Keeping it simple, that would mean a 20/31416 chance every week or 1/1571.

Let us say we want to be 75% sure that we eventually get that Lamborghini Urus S. That is the same as saying that we only lose in 25% of the times we play a series of weeks.

The possibility of losing one time is 1570/1571. The possibility of a series of events is found by multiplying the probability of each event in the series.
The question becomes "how many times do you have to multiply the (very close to 1 possibility) of losing to reach 0.25 or less?".

  • (1570/1571)^n ≤ 0.25.

Now, we can solve for n:

  • n ≥ log(0.25) / log(1570/1571)

Using a calculator, we find: n ≥ 5733. So after you have played a series of 5733 weeks of 20 tickets in Zone 1 and 2 you would have a 75% probability that you won at some point.

Given 52 weeks in a year you have to play for 110 years but if you play 100 tickets every week you can get down to 22 years.
The price of those tickets would have been (7854/31416) * 5733 * 20 * £ 5.79 * 0% + ((31416-7854)/31416) * 5733 * 20 * £ 5.79 * 50% = £248,955.

Wait, what!? that is more than the car cost! Why is that a favourable bet?

Well, we wanted 75% assurance that we would win in the span of those plays.
If we settle for 50/50 chance we can reduce the number to 1088 weeks of 20 tickets which is (only) £47,246 over 21 years.
Play 40 tickets and you half the time. Play 80 and you half it again.

So if you are a good player and you really go to town with your ticket amount there is a good chance you will be winning at 4-5 year intervals.
That is why there are some multiple winners and some that play a lifetime without ever winning anything

It follows along the same kind of theory that some stockbroker will have an incredible winning streak 30 years running. This is not down to his God-like ability to pick stocks but rather that given a large enough sample of stockbrokers some are going to perform through the roof based on randomness alone.

You can now take a look at your result history and calculate what you average accuracy is.
Plug in the amount of tickets you play and what certainty you want at winning and you can then see what it would cost and how long it would take. If you play well enough it is actually not a bad proposition compared to betting on the stock market for instance ..

Let me point out that the above numbers are based on completely random distribution of your tickets in zone 1 and 2. There are still some tricks that can get you an even higher probability at a smaller cost but that is a topic for another time.

My story

Back when everyone was in pandemic lock-down and we were sitting unshaved at home working in our underwear I took a look at these numbers. I couldn't believe it.

But knowing that assessing bets with my gut feeling wasn't accurate I decided to trust the numbers and go for broke.
I put together a small pool of money for the experiment (around £1,000 I think it was) that I decided to just pour into BOTB and play maximum tickets every week or at least the weeks where I could sense some sort of potential for consistency.

To optimize my chances I decided on the less crowded Midweek competition to reduce my risk of tie-break and I made a little application that could plot my 100-150 tickets every week with 1 pixel spacing in a chess board like pattern so I wouldn't have to sit and try to enter them manually while my arm was cramping up.

I settled for a ticket price that aligned with my stats and how many times I had to play in order to have a good chance of winning
(I will point out that the image resolution and zone sizes have changed markedly since then - apparently without any fuss or commotion).

Cars in the right price bracket for my pool of money were the Mercedes A45 S and the Porsche Macan.
The week my little pool of experimental money ran out I received the call from Christian - 2 months into my experiment.
The bet had paid off. I was thrilled but looking back half of the thrill was actually the sense of satisfaction that my analysis had been accurate.

They say that a sign of insanity is to do the same thing over and over and expect a different outcome.
I guess they don't understand how randomness and statistics work ...