In the UK National Lottery 6 numbers (plus a 'bonus ball') are drawn from the set of 1-59.

Under normal circumstances, in order to win the jackpot the six numbers on your ticket must match those drawn. If no one matches those six numbers the jackpot rolls over to the next draw.

Because the jackpot has reached a certain amount (£57.8 million) tomorrow's draw has a must-win condition. This means that if no one matches the six numbers then the jackpot will go to anyone meeting the criteria of the next tier (e.g. matching 5 numbers and the bonus ball), or the tier after that if again there is no winner, and so on. The tiers and probabilities are as follows:

Match 6                 1 in 45,057,474
Match 5 + Bonus Ball    1 in 7,509,579
Match 5                 1 in 144,415
Match 4                 1 in 2,180
Match 3                 1 in 97
Match 2                 1 in 10.3

Clearly a must-win condition increases a ticket's chance of winning (at least part of) the jackpot.

However, there is a second way to become a millionaire: Each ticket also has an entry in to a 'millionaire raffle' in which 1 of the tickets bought will win £1 million, and the chance of winning is:

1 in # tickets bought

Because of the must-win condition (and huge jackpot) the number of tickets sold (and therefore number of raffle entries) will be greatly increased, meaning a ticket's chance of winning the raffle are decreased.

This leaves a dilemma: Will my overall chances of becoming a millionaire be greater in tomorrow's draw due to the must-win jackpot or lower because increased raffle entries?

For what ranges of # tickets sold are your chances of winning either the jackpot or raffle equal when there is a must-win condition v.s. when there isn't? i.e. How many fewer tickets would have to be sold in a normal draw to match the chances in a must-win draw.

  • Assume each ticket chooses unique numbers (bonus: what if tickets use non-unique numbers?)
  • Ignore the fact that sharing may mean winning the jackpot won't make me a millionaire

Trying to answer my own question. Warning: I'm not a mathematician.

It occurred to me that chance of winning the jackpot in a must-win scenario would be 1 in # tickets bought, the same as the raffle.

Therefore, combining the probabilities of a jackpot or raffle win gives us the # tickets required in each draw to give equal odds in both scenarios:

a is # tickets in non must-win draw
b is # tickets in must-win draw
(in ~millions)

1/45 + 1/a - 1/45 * 1/a = 1/b + 1/b - 1/b * 1/b

(a + 44)/45a = (2b - 1)/b^2

e.g. for a non must-win draw with 25 million tickets bought:

(25 + 44)/(45 * 25) = (2b - 1)/b^2
b = 32.10078

Meaning a must-win draw would have worse odds if ~32 million or more tickets were bought.

Unfortunately (even if I got that right) it still doesn't help us with the real world scenario where many of the tickets don't have unique numbers.


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