Misleading Stats

Earlier this month the BBC reported this story Second lottery win for US woman.
The woman had won the $1m jackpot on two scratch cards in the last 4 years.
The odds of winning the first card were 1 in 5.2 million, whilst the odds of winning the second card was 1 in 705,600, so experts claim that the chances of her winning both games were an incredible 1 in 3,669,120,000,000. However these experts (or BBC researchers) are wrong. Yes if she had walked in to her newsagents and said can I have one Cool Million scratch card and can I also have a Jubilee one too, then the odds would have been that high. However I pretty sure that she will have purchased more than one of each of the scratch card, and for each time she purchase another one from the same game he odd drop.
For example the odds of winning a small lottery which only has 10 outcomes is 1 in 10. However if I purchase two different tickets for the same draw my odds half to 1 in 5. If I purchased all 10 combinations I must win. And the same goes for this woman. Lets assume that she loves scratch cards and that she had purchased 200 of each before she won the jackpot. Her odds would change from 1 in 3,669,120,000,000 to 40,000 in 3,669,120,000,000 or 1 in 91,728,000 which is a slightly smaller number (but still impressive).

3 thoughts on “Misleading Stats

  1. Hmmm.

    I think the questions is rather different. No one (except herself) was interested in the odds that she in particular was going to win again. The probability we are really measuring is whether ANYONE who had already won more than a million would win a million again (anyone of the previous winners winning again would generate a news story)

    The odds that one winner would win again are really quite low, in my view.

    There is a familiar problem with birthdays that is relevant. If you are in a room with 30 people it is unlikey that you have the same birthday as anyone else there. But it is likely that two people in the room share the same birthday.

  2. I’m not sure you can multiply up like that for scratchcards the way you would for a lottery.

    If the lottery has ten possible outcomes, then buying tickets covering all ten guarantees a win. But ten independent scratchcards each have the same odds of winning, independent of each other – you don’t increase your chances tenfold.

  3. I should add that you’re right if that’s how their “lottery scratchcards” work – i.e., there is a winning card, like a golden ticket. In that case, buying more cards does increase your chances.

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