Lotteries

Lotteries are, in EV terms, a bad gamble (usually). In terms of winning an obscenely large amount of money for a £1 or £2 outlay, however, there is nothing to compare. But which lottery gives the best return to the player, and which gives the most to charity?

To start with, a comparison between Lotto (the 1994 "original" draw, part of the National Lottery family of games) and The Health Lottery (the 2011 newcomer, promoted heavily by Richard Desmond's Northern and Shell group).

Lotto

Tickets cost £1.00. Choose 6 numbers from 49. 6 main numbers are drawn, plus 1 Bonus Ball number. Win a prize if you match 3 or more main numbers.
Match 3: 1 in 57, win £10
Match 4: 1 in 1,032, win £62*
Match 5: 1 in 55,491, win £1,518*
Match 5 + Bonus Ball: 1 in 2,330,636, win £101,987*
Match 6: 1 in 13,983,816: win £1,988,749*
Win any prize: 1 in 54

* prizes for each tier are allocated after all £10 wins are accounted for, and are a percentage of the remaining prizepool. Figures assume each number combination is bought once and exactly once, and there is no rollover or other special prize.

Health Lottery

Tickets cost £1.00. Choose 5 numbers from 50. 5 numbers are drawn. Win a prize if you match 3 or more main numbers.
Match 3: 1 in 214, win £50
Match 4: 1 in 9,900, win £500
Match 5: 1 in 2,118,760, win £100,000
Win any prize: 1 in 209

So if you buy 1 ticket a week, then it will be a little over a year between wins with Lotto, compared to 4 years between wins with Health Lottery.

Charity: 28% of the purchase price of a Lotto ticket goes to a good causes fund, compared to 20.3% of the purchase price of a Health Lottery ticket.

Payouts: 45% of the purchase price of a Lotto ticket is paid out in prizes during a normal draw. The Health Lottery is claiming that 57% of the purchase price of their ticket is paid out in prizes, however given the parameters of the game and the odds of winning (which are unintuitively placed on their website, and confirmed by combinatorial maths) just 33.4% is paid out in prizes. I have twice emailed The Health Lottery about exactly what the prize breakdowns and where the ticket price goes to are, yet despite the autoreply saying I should expect a reply by the following evening, have been waiting for 2 days and 9 days for a reply.

Summary: The Health Lottery gives away less to charity, and on an average draw (as worked out from the odds given on their website) gives less back to the players. What is more, it takes four times longer for the player to see any sort of return (on average). Add to this the frankly terrible customer support being offered by the Health Lottery and it's very hard to see why anyone would play it. It will be interesting to see how the Health Lottery plans to reconcile the 57% it claims gets paid out, with the 33% that actually gets paid out.