If you haven’t already please read Cassava Bingo Part 1 otherwise this won’t make much sense!
Consider doing 37 offers with a £10 deposit and a 300% bingo bonus e.g. Bingo Ballroom:
Let’s say we wager the £370 by placing £10 on red or another even money bet each time. On average 18 of these will win giving us a net of 18*£20 – £370 = £-10. Of the 19 that lose however we will gain a bingo bonus each time. The total bonus over the 37 offers will be 19*£30 = £570. In other words we are risking a £10 cash loss for £570 of bingo tickets.
If we place our £10 on a single number we would expect it to win once every 37 spins. Our net loss will again be £10 since we would receive £360 if the single number bet wins. However on the 36 other occasions we lose we will receive a bingo bonus totalling 36*£30 = £1080, nearly double the amount of bingo tickets for the same expected loss.
Now let’s calculate the expected value of the bingo bonus if placed on Funky Monkey tickets. Both theoretically and experimentally the likelihood of a 1LN, 2LN and FH win are all approximately the same. From looking at my bingo statistics the likelihood of 2 people winning 1LN, 2LN and FH is about 25 – 33% of one person winning (let’s say 33% to be conservative), and three people winning about a sixteenth of 1 person winning. Ignoring the chance of three people winning since it is small, this means that given a win the rough probability of each type of win is:
1LN: 25% (¼)
2LN: 25% (¼)
FH: 25% (¼)
1LN (2 people): 8.33% (1/12)
2LN (2 people): 8.33% (1/12)
FH (2 people): 8.33% (1/12)
Now we also know that the RTP on the low end is approximately 50% and that the 1LN, 2LN and FH wins are
£50/£150/£300 so using this information let’s consider the scenario of wagering £3500 worth of tickets. This gives us a theoretical total win of £1750. Given the probability distribution of each type of win we just estimated:
1LN: 3 wins = 3*£50 = £150
2LN: 3 wins = 3*£150 = £450
FH: 3 wins = 3*£300 = £900
1LN (2 people): 1 win = ½*£50 = £25
2LN (2 people): 1 win = ½*£150 = £75
FH (2 people): 1 win = ½*£300 = £150
Total = £1750
We can now calculate the expected value of the bingo bonus. In this instance we’ll continue with the assumption that we deposited £10 for a £30 bingo bonus and we now have £90 left to wager on smaller bingo rooms (we needed to wager 4*£30 = £120 and we have just wagered £30).
1LN: £90 wagered expects £45 (50%) to be lost. Since we only won £50 we are left with £5 and since we are expected to win 1LN by ourselves 3 times every £3500 this gives an EV of £15 per £3500 from this component.
2LN: 150-45 = 105. 105*3 = £315
FH: 300-45 = 255. 255*3 = £765
1LN (2 people): 25 – 45 = -20 i.e. this is zero since we bust out with our bonus
2LN (2 people): 75 – 45 = £30.
FH (2 people): 150 – 45 = £105.
Total: £1230 profit from £3500 of bingo tickets after wagering is completed (~35% yield).
So let’s compare our two examples with different roulette bets from a while back:
After 37 games betting £10 each time on red we expect to lose £10 and gain £570 worth of bingo bonuses. After wagering we expect a 35% yield i.e. ~£200 of winnings from bingo. This gives us a net profit of £190 after 37 offers or ~£5/offer.
After 37 games betting £10 each time on a single number we expect to lose £10 and gain £1080 worth of bingo bonuses. After wagering we expect a 35% yield i.e. ~£380 of winnings from bingo. This gives us a net profit of £370 after 37 offers or ~£10/offer.
I have created a graph to demonstrate this more clearly:
Now it’s all well and good being able to double our expected profit but I imagine most people aren’t willing to place £10 on a single number roulette bet as the variance is going to be far greater than an even money bet. There’s no point of having greater EV if we are more likely to lose our bankroll from variance! And believe me my biggest single number bet losing streak is 189…
Now I could bore you with the statistics of variance/standard deviation (σ) however it’s not necessary and you only need to know that it represents the spread of how much we can win or lose from our expected value. I’ve plotted this against roulette numbers covered again.
The graph shows that despite doubling our expected profit, a single number roulette bet increases our standard deviation by nearly 6 times compared to an even money bet! To give an idea of how large this is, consider the following two sets of data representing hypothetical profit and loss:
Set 1: 1, 3, 5,7 ,9
Set 2: -19, -7, 5, 17, 29
The average profit for both groups is 5, however the standard deviation is 6 times as great (σ1 = 2.82 vs σ2 = 16.97).
For those who don’t want to have a large bank to avoid lose their bankroll we can optimise our risk adjusted return by optimising the Sharpe Ratio defined as EV/σ.
A similar shaped curve is produced for all bingo bonus amounts until you get to around a 50% bonus, which I don’t recommend doing as it’s barely profitable. The take home is that as far Risk adjusted return is concerned a bet on red/black is best, however a 40% increase in profits can be obtained with a column bet with a 40% increase in σ, which is very low to begin with.
Suggested Starting Bankroll
I wrote a quick simulation in Octave/Matlab to see what the expected drawdown of each method is. A thousand trials were run each consisting of a thousand bets. The code for the even money bet is shown below:
clear() clc() for j = 1:1000 for i = 2:1001 x(1) = 0; a(i) = rand(); %Roulette spin b(i) = rand(); %Bingo if a(i) < 18/37 x(i) = x(i-1) + 10; elseif a(i) > 18/37 if b(i) < 3/116.67 x(i) = x(i-1) + 5 - 10; %1LN elseif 3/116.67 < b(i) && b(i) < 6/116.67 x(i) = x(i-1) + 105 - 10; %2LN elseif 6/116.67 < b(i) && b(i) < 9/116.67 x(i) = x(i-1) + 255 - 10; %FH win elseif 9/116.67 < b(i) && b(i) < 10/116.67 x(i) = x(i-1) - 10; %1LN 2ppl elseif 10/116.67 < b(i) && b(i) < 11/116.67 x(i) = x(i-1) + 30 - 10; %2LN 2ppl elseif 11/116.67 < b(i) && b(i) < 12/116.67 x(i) = x(i-1) + 105 - 10; %FH 2ppl elseif b(i) > 12/116.67 x(i) = x(i-1) - 10; endif endif endfor lowest(j) = min(x); j endfor bust = min(lowest);
Here are the results:
|Numbers covered:||Loss > £50||Loss > £100||Loss > £200||Loss > £400||Loss > £800||Max Loss from all 1000 trials (£)|
If we assume we want the chance of losing our starting bankroll to be less than 1% then the suggested starting bankroll for even money, column and single number bets is around £140, £200 and £700 respectively based off the results.