please empty your brain below

Thanks for doing the stats, DG. So, the odds on the Conalot lottery are becoming even more atrocious. But the many cash-strapped people who could put their stake to better use are sadly unaware of this.

Disclaimer: Although I've never played the lottery, I do have the absolute *minimum* number of premium bonds, just so that as an irrational human I can entertain the tiny illusion (and miniscule chances) of a life-changing win :-) At least I can get my money back.

"Posted at 00:59" - yes we do notice!
"Expected winnings" is a technical term in statistics, and you're not using it properly. You seem to be using it as the most likely winnings, whereas it should be the amount you would win if you won multiplied by the probability of you winning it. For example, your expected winnings from matching four balls from playing the lottery for a year before they change it are £2.39, not £0 -- 100 x 1/144415 x 52.

Calculating expected winnings from the jackpot and the raffle is actually quite difficult, because you have to make assumptions about how many other people are playing the lottery (and to be truly accurate, how many of them choose their own numbers instead of using lucky dip, and how good they are at choosing random numbers (not very)).
Hi Mike.

Yes, I know I'm not calculating 'expected winnings' properly (see start of last paragraph), because I'm ignoring fractional returns from the more unlikely prizes.

But in reality, as a single punter you're not going to win those fractional amounts, so I hope my figures are a more realistic interpretation.
I have read a calculation as to the mathematically best time to buy your lottery ticket.

That is, (or was) sometime after Friday evening when the odds of you being knocked down and killed by a car fall below the odds of you winning the lottery, thus giving you the opportunity to be alive to enjoy your winnings....
There is a certain intuitive simplicity to DG's statistics. Fortunately, or perhaps unfortunately, real statistics needs to deal with low probability, high impact events. DG's approach would suggest, for example, that you should forget about paying insurance premiums (what is the likelihood of actually making a claim in any one year, or decade?) and enjoy a nice holiday instead. Loss aversion is part of human nature, over-valuing downside costs and under-valuing upside benefits.

That said, the lottery is clearly a waste of money, even if you take into account the small change of winning one of the larger prizes :)
To increase my chances of winning a raffle I always put both halves of the ticket into the draw. Is there an equivalent for the National Lottery?
Technically, over a 10 year period your expected winnings from the 5 ball prize are £7.96.

Ignoring this (because winning is a once in 1250 years event) makes me a bad mathematician, and a useless insurance salesman, but hopefully a decent realist.
Andrew's point about insurance premiums is right, although not perhaps in the way he meant. Insurance companies make fat profits because most people never need to claim. It is just a reverse lottery, with added fear factor. The only insurance I have is that which I am legally or contractually obliged to take out (buildings for the mortgage, car, boat) and always at the minimum possible level. This isn't actually as reckless/bold as it sounds as the minimum required includes third party (and, for the boat, salvage), which is where any big bills are likely to come from.
I have never wanted to indulge in the lottery, and always bring it home to myself that choosing the numbers 1, 2, 3, 4, 5 & 6 has just as much chance of winning as any other combination. Even so, I would instinctively be loath to choose those numbers, as most others would.
Actually, if most people would be loth to choose 1,2,3,4,5,6; bucking the trend by doing so would increase your chance of not having to share the jackpot. Other counterintuitive strategies, like choosing the numbers that won last time, have a similar perverse effect.
Of course, they don't increase your chance of winning - just the size of the prize if you do win.
My own lottery record is break-even. Zero in, zero out.
Such analysis should be in the public domain, but will any of the most-read or watched media outlets publish it - NO.

The lottery is (and has always been) a form of voluntary taxation for the less-well-informed in society.
"£550 in winnings might sound good, but over ten years you'd have paid out £2088 for your tickets"

As you said, NOT a good investment. If you bought more than one ticket, and many do, you'd have wasted a whole lot more. Hmmm, and they say there are people living below the poverty line. Do they play the Lottery? Cos that's a sure way of staying there.
Sorry for this, but to quote Cancer Research's website:

"One in two people born after 1960 in the UK will be diagnosed with some form of cancer during their lifetime"

We don't tend to dwell on that one, but imagine the odds of winning the lottery was 1 in 2 over your lifetime we'd all be queuing up!!!
I remember when the lottery launched, as I was at school. My business studies teacher described it as a tax on the stupid. I don't think it is far from the truth!
I think you guys are under-estimating the common people's intelligence. However, sheer sense of greed (or as you put it -- "Someone is going to win it") is already enough for some well informed bloke to keep playing.

Lottery wouldn't die even among a sufficiently enlightened population, which I believe the UK has.
DG is calculating modal values, not mean values, nothing wrong with that, the data distributions will be such a weird shape that it's not clear to me that any other system would make more sense.

In order to see that modal values make perfectly good sense, compare the number of people at or near the modal value (never major prize winners) with the number of people distant from the modal value (who win major prizes).

The insurance example is a bit of a red herring because there are a lot more people involved in fires and road accidents than national lottery winnings.
And, of course, the losing value for fire or accident isn't £0, it's all your worldly goods and becoming bankrupt or even destitute. Some people don't notice that not all risks are covered by insurance policies anyway. Last time I looked, all travel insurance policies excluded anything to do with terrorist attacks.
Your calculations are close enough for me, and it intuitively feels like a much greater scam that is being introduced.

There's also that thing in negotiating about 'giving away something that has low cost to you but a higher perceived value to the other person'.

Hence the rather cynical introduction of 'winning a free go'.
1,2,3,4,5,6 would actually be a terrible combo to choose if you were trying to avoid sharing the prize: it is the most commonly picked set of numbers (around 10,000 per week). You would be very, very disappointed with your jackpot if those numbers ever came up
@Max Roberts
"the data distributions will be such a weird shape"
Dredging up A level statistics, isn't is a Poisson distribution? (large number of trials with very low probability of any individual one being a success)?
A mathematician will tell you that if you really must play the lottery, make sure you just buy one ticket. That way you minimize your expected loss. Think of it this way: if you bought all the tickets you'd win every prize -- and maximize your loss!
@ Jon Combe

It is a tax...a tax on the poor. Only good thing is can choose not to pay it.
To: SJM. As someone who has lost four close aunts and uncles,two grandparents,a fatherand a brother to cancer; a husband and a son still undergoing treatment for their cancers; I think that the odds now might be a little more favourable to me in the lottery than your stated one in two. I will go forth tomorrow and buy a ticket!!
Timbo, these data have such a massive positive skew that it's not clear to me that parametric statistics have anything meaningful to say.
I agree that the lottery is partly a tax on the stupid. But not entirely; I know some people who are certainly not stupid, who do understand the odds, and who do regularly buy lottery tickets.

The way I explain this is that anyone can put whatever value they like on the dream of a life-changing win. Myself I value that dream at the mathematical expectation figure, therefore I do not buy a ticket. DG evidently values the dream even lower, at £0, which is his right. And my non-stupid friends apparently value it higher than their mathematically expected loss - which is their right.
Agree what has been said re tax on people etc. The reason I have no doubt people will keep playing is the odds of winning the US Powerball is 175 million to 1 yet people still go crazy there when the jackpot is a few hundred million dollars.
and what about the great projects that are supported by the National Lottery ? Great that many people buy tickets - great for those who win, sour grapes from those who don't.....
I joined in with a syndicate in the office. I know it's a waste of money BUT I can't stop because I know if I do my colleagues will win and I'll be the only one left at work.

My only way out is to change job and sever all ties with my current workmates.
I never realised how thick I might be until I tried to understand the analyses above. Yes, the Lottery is clearly a con (for players, not for the 'good causes'), odds against winning now even worse. On top of that, you never know how many others might share 'your' jackpot.

I don't do the Lottery but I do have some Premium Bonds, enough to allegedly win maybe every four months, based on my limited understanding of raw numbers.

But - how much 'cleverer' is it to bet on PBs? The odds of any one Bond coming up each month is apparently 1 in 26,000 - and the most likely win is £25 (a lot of those in the draws) - so that's maybe a 3 x £25 'return' annually, still worse than any of the High Street crooks who borrow my money and pay it back at lesser value still.

Makes bank robbery (with the associated severe risks) seem a better bet. Anyone got a spare laser drill?










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