Would You Win More $$ If Toto Dropped to 5 Numbers?

 

If you haven't been living under a rock recently, you would've came across the new year draw featuring a massive 8.2 million prize pool. It was supposed to be just 5 million but since the past 2 draws had no group 1 winner, the prize pool has snowballed to be worth more than 10 HDB flats. 

Given that Toto has featured the drawing of 5 numbers during it's inception back in the 1960s, I was thinking whether it would be better for both consumer and operators if we adjust it to draw 5 numbers but from a bigger pool of numbers. But before we delve into that, for the unculturedinitiated...

Just What is Toto?

Toto is a popular lottery game in Singapore operated by Singapore Pools. Players choose at least six numbers from 1 to 49. During the draw, six main numbers and one additional "bonus" number are drawn. To win, players need to match at least three of their chosen numbers with the drawn numbers.

Key Rules:

• Minimum Bet: $1 per set of six numbers.
• Winning Combinations: There are multiple prize tiers, with the jackpot (Group 1) going to those who match all six numbers. Lower prizes are awarded for matching 3 to 5 numbers, with or without the bonus number.
• System Entries: Players can increase their chances by betting on more than six numbers in one ticket, forming multiple combinations.
• Draw Days: Held every Monday and Thursday, with special draws occasionally.

That's the general gist of the Toto and it's rules that I asked ChatGPT to type out. You may access the full list of rules here.

So What Are The Odds of Winning?

TLDR: It's 1 in 14 million

For those who have been listening to their E-Maths lesson, you would probably be able to easily calculate the odds. But for those who haven't been paying attention to their Maths teacher back then (shame on you), here's some math so bear with me for the moment.

In general, the mathematics behind hitting the Group 1 Jackpot in Toto (matching all 6 numbers) relies on the concept of combinations. Here's the visual representation of the combination formula:

And now for some quick maths, we just need to substitute n for 49 and r for 6. It would look something like this:

Now if you type this into your phone calculator, you would be able to compute 10,068,347,520/720 = 13,983,816. You are not mistaken, your odds of winning Group 1 in Toto is 1 in 14 million. So my advice to you is to stop donating to building the next national stadium. But for those 没买没希望 degens, thanks for contributing back to our society o7

If you are interested, I went ahead to calculate the odds of winning each prize group below:

1. Group 1 (Jackpot): Match all 6 numbers

   • Odds: 1 in 13,983,816

2. Group 2: Match 5 numbers + the Bonus number

   • Odds: 1 in 2,330,636

3. Group 3: Match 5 numbers

   • Odds: 1 in 55,491

4. Group 4: Match 4 numbers + the Bonus number

   • Odds: 1 in 22,197

5. Group 5: Match 4 numbers

   • Odds: 1 in 1,083

6. Group 6: Match 3 numbers + the Bonus number

   • Odds: 1 in 812

7. Group 7: Match 3 numbers

   • Odds: 1 in 61

Overall Odds:

• The overall odds of winning any prize in Toto are approximately 1 in 54.

Now What If We Dropped Toto To Just 5 Numbers?

I know what you are thinking. If we drop to just picking 5 number obviously the odds will increase right? Of course, but if we were to continue along with our experiments of dropping the number of balls being picked (yes, we still pick out the number balls manually), then we would also have to adjust the total number of balls inside the machine.

In fact, this was just a random shower thought that sprouted out so I went ahead to google whether there were lotteries out there that selected 5 numbers for the jackpot. Apparently, most gaming operators out there have the same idea in mind where a jackpot required 6 matches. 

The Euro Millions and Lotto from the UK's National Lottery

The game that is closest to Singapore's Toto when dropped to just 5 numbers would be the Euro Millions Hot Picks where choose 5 numbers from 1 to 50 and need to match 5 numbers for the jackpot so your odds is just 1 in 2,118,760. 

Before you close this window, you need to remember that this game needs to be fair for both the operator and participant. Pools can't be operating at a loss right, if not how to generate 10 billion of revenue in 2024?

As such, we would want to increase the number of balls being picked to be the same as those of the UK national lottery's Lotto. The Lotto allows player to pick six numbers from 1 to 59 where you need to match all 6 for the jackpot. We will be taking the same approach in our modified version of our Toto featuring 5 numbers picked.

So What Are The Odds of Winning Toto with 5 Numbers Drawn?

TLDR: Don't have cos I want you to read finish (takes effort to write one hor)

If you remember our combination formula earlier (as seen below), then you would know that we just need to substitute certain numbers to figure out the odds of the updated Toto

Here's the updated game configuration:

1. Number Pool: 1 to 59
2. Numbers to Match for Jackpot: 5

Now to calculate the odds, we just need to substitute n with 59 and r with 5. Here's a visual representation of the calculation:

And the final results is that the odds of hitting the jackpot is 1 in 5 million.

Since the odds are better, naturally there will be more Singaporeans being enabled to throw more money for that little piece of hope while in theory would lead to more revenue for Singapore Pools. 

So why has Pools not improved the odds over the years? We need to remember the original mission of the organisation which can be seen from the screengrab I've taken from the press release back in 2007 from Sports SG.

Pools doesn't operate as an organisation to maximise earnings but rather is an outlet for people to buy a little piece of hope. In fact, they have contributed more than $2 billion to nation building (together with others from the Tote Board Group). So next time you walk by an outlet while going back to office from lunch, feel free to make that $2 donation to our nation building.

And there we have it, the odds of our adjusted Toto game with 59 numbers and 5 balls drawn would increase our odds of hitting the jackpot from 1 in 14 million to 1 in 5 million.

P.S Initially there was a simulation section where I used python to see how many tickets it would take before I get a winning ticket under the original Toto rules. However, after running for more than 30 mins there was no winning ticket generated so I went to google and realised that it would take too long to generate 14 million iterations so I promptly closed the programme.