In a departure from our usual subject matter on Sofabet, I recently found myself discussing the wisdom of holding Premium Bonds: do they, the National Lottery or Euromillions give the best chance of a life-changing big win? I couldn’t find a recent calculation online so attempted to crunch the numbers for myself, and am posting the calculations here in case others also find them interesting.

My conclusion surprised me. By my reckoning, it’s not even close – it’s the Euromillions, specifically on a Tuesday.

Before getting into the figures, it’s fair to note that Premium Bonds seem to be a very different beast from lotteries. Not for nothing are the latter known as a “tax on stupidity” – no more than half of ticket sale revenue comes back as prizes, and as it’s a random draw there’s no way to tilt the odds in your favour. To put those odds in context, if you buy a ticket from a shop seven miles away, you’re more likely to be killed driving home than to win the jackpot.

Premium Bonds feel like a much more respectable investment, because you can always sell them for what you paid for them. But they, too, involve a financial sacrifice, because your expected winnings are less than the interest you could get in a savings account. So if you’re willing to sacrifice some cash for the chance of a life-changing win – and let’s arbitrarily define “life-changing” as a million plus – it’s meaningful to consider them together.

Let’s start with Premium Bonds. The annual prize fund is currently equivalent to 1.3% interest on the total amount held in bonds. Let’s say, just for the sake of this calculation, that you hold the maximum allowance of £30,000. On average, over a year you’d therefore expect to win 1.3% of that in prizes, or £390.

(Of course, in reality, you’d almost certainly get less than £390, because that figure is skewed by the existence of the big prizes – a few will get a lot more, the vast majority will get a bit less. But it’s in the ballpark, so let’s stick with it for the sake of simplicity).

The best interest rate you could get in a bank is about 3%, meaning your £30,000 would earn £900 at the end of the year. Assume you’re a basic rate taxpayer, and you have to pay £180 of that in tax, reducing your interest to £720. So, effectively, keeping your £30,000 in Premium Bonds instead of the bank is costing you (£720 minus £390 =) £330 a year.

What do you get for your sacrifice of £330 a year? Your 30,000 bonds entered into 12 monthly draws, in which each bond has a 1 in 45,527,700,191 chance of winning a million. That equates to around a 1 in 126,466 chance of winning a million in the course of year.

What if you sold your Premium Bonds, put the money in a savings account, and instead said that you were willing to lose £330 a year on the National Lottery?

With the Lotto, 45% of ticket sales goes into the prize fund – you expect to lose 55% of your stake. So, for an expected loss of £330 a year, you would be looking at buying (330 x 100/55 =) £600 of tickets.

(In reality if you spent £600 on Lotto tickets you’d almost certainly lose more than £330, because – as with Premium Bonds – this “expected loss” figure is skewed by the existence of big prizes: most will lose more, a few will win big. But, again, it’s a ballpark figure so let’s stick with it for the sake of simplicity and consistency).

In October, the price of a Lottery ticket goes up to £2, so £600 would buy you 300 tickets over the course of the year. Each of those tickets has a 1 in 13,983,815 chance of winning the jackpot. That comes to a chance of winning a jackpot, from your willingness to lose £330, of about 1 in 46,546.

To summarise so far: I reckon that, given current interest rates, pound-for-pound you’re nearly three times as likely to win a million by playing the Lotto as you are by buying Premium Bonds – even accounting for the fact that you can always sell your Premium Bonds, whereas a losing Lotto ticket is gone forever. That surprised me.

What about the Euromillions? Here, 50% of ticket sales go into the prize fund, meaning you’d spend £660 a year for an expected loss of £330. At £2 a pop that’s 330 tickets, each of which have a 1 in 116,531,800 chance of winning the big prize. And that equates to a chance of winning a jackpot, for the expected sacrifice of £330 a year, of about 1 in 353,126.

This doesn’t look good. It’s worse odds than the Premium Bonds, and much worse than the Lotto.

However, we also have to factor in the “Millionaire’s Raffle” element of Euromillions, through which one ticket bought in the UK is guaranteed to win a million in each draw. The chance of winning this obviously depends on how many people in the UK buy tickets, which according to the Euromillions website is about 3,500,000 on Tuesdays and 9,200,000 on Fridays.

Once you factor this in, I reckon that if you bought all your 330 tickets on the Friday draws, you’d have about a 1 in 25,839 chance of winning at least a million, either from the Millionaire’s Raffle or the main jackpot. If you buy them all for the Tuesday draws, however, those odds improve to 1 in 10,296.

The conclusion, then? If you’re the kind of person who’s willing to write off some cash in return for gaining the remote possibility of a really big win, then by my calculations you’d be giving yourself a better chance if you sell your Premium Bonds, put the money in the bank and spend some of the interest on the Tuesday Euromillions instead. I make it about twelve times as likely to win you a million.

If you spot a flaw in these calculations, please do correct me in the comments below.

I’ve spotted a flaw straight away!! I haven’t got £30000 dam!!

Great article by the way, looking forward to X Factor!!

🙂 Cheers, Shoulders, us too!

Just thinking of selling my premium bonds actually, having come to the conclusion they’re an absolute waste of space. My Dad keep getting the odd £20. I haven’t won a penny.

I won’t be plowing my funds into the Euromillions though, more like investing in something fast and petrol-hungry.

Hi, love your blog, especially eurovision stuff.

if you had a choice of 10 euromillions tickets for 1 tuesday or 1 ticket each week for 10 tuesdays in a row, which is more valuable? i assume its the 10 tuesdays in a row.

Hey Colm, welcome to Sofabet! Interesting question – my first thought was that it must be the 10 for the same draw, on the grounds that (assuming of course you choose different numbers) each ticket has a progressively slightly better chance of getting the jackpot – so if the first one loses, the second one has a 1 in 116,531,799 chance, the third has a 1 in 116,531,798 chance, etc. Cumulatively that’s slightly better odds than one for each of ten Tuesdays, each having the 1 in 116,531,800 chance.

But then I realised it’s a different equation with the Millionaire’s Raffle, given that here the odds depend on the number of tickets bought. If you assume exactly 3,500,000 other tickets are bought every week, then buying 10 for one draw would reduce the chances of your first ticket winning to 1 in 3,500,010 (the second 1 in 3,500,009, etc.). Whereas one every Tuesday for 10 weeks would each have a 1 in 3,500,001 chance.

I reckon the slightly better odds for the Millionaire’s Raffle must outweigh the slightly worse odds for the Euromillions jackpot, meaning that you’re right, it’s the 10 Tuesdays in a row. Happy to be corrected by readers whose brains hurt less than mine does when pondering this kind of question.

[Lotteries] “… and as it’s a random draw there’s no way to tilt the odds in your favour.”

It is rumoured that picking numbers that are not days of the month can increase your chances of not sharing a jackpot or other prize as many players pick numbers on the basis of dates they find significant.

On another aspect, I’d also guess that further to the Tuesday/Friday split on the millionaire’s raffle, there would be a noticable difference between the number of tickets bought on rollover and non-rollover weeks.

Maybe, to maximise the millionaire’s raffle chances, it’s best to buy all the tickets on a Tuesday directly after a multi-week rollover was won on a Friday?

These are both great points Danny. You would think that in general higher numbers will give you a better chance of not sharing jackpots – above 31 can’t be days of the month, above 12 can’t be months of the year, house numbers will skew lower as many streets have fewer than 49 houses, etc.

Actually there’s research on this – better, too, it seems, to avoid numbers that make pretty patterns on the slip –

http://www.hpcc.ecs.soton.ac.uk/lottery/201.html

http://www.southampton.ac.uk/~sjc/research/research012.htm

Wow, the conclusion that you could potentially double your winnings (second link) by picking unloved numbers is a real eye opener!

Just wanted to post to say that I think this is a brilliant article. Really enjoyed it and might buy the odd Tuesday euromillions ticket now.

Presumably the value is actually in buying a ticket when the jackpot is lower, as less people will play and there is more chance of landing the millionaires raffle?

Also, you state that the odds of winning big on the lotto, once the stake increases to £2, are about 1 in 46,546. Would this be 1 in 23,273 with the current stake of £1?

Cheers Mrs Shrewd! Yep, I think that’s right, both about the odds with the current £1 stake and the assumption that fewer people will play when jackpots are lower.

Maybe it is true that if there is an opportunity to win £1 million on multiple days in the week, the best chance to win at least £1 million (or the jackpot which will presumably always be above £1 million) will be improved if you play on Tuesday, which is presumably the day on which there are least players.

The numbers are very suspect though, and the best chance of winning is probably the multiple chances of winning £1 million I understand you can win on occasion in the UK lottery or in the Euromillions.

Whatever lottery you play, it is my assumption you will increase your chances of winning (ever so slightly) by using the same numbers, reason is the lottery is completely random, given that thought, thunder and lightning are also random, if you were to stand in the same place in a field, and you could live forever, you would eventually be struck by lightning, it might take a thousand years or more, or it could happen tomorrow, so imagine you are one of the same numbers picked every draw (let’s put a number 5 sticker on your head) I hope you get what I’m stating here, pick your number set and stick with it.