California Lottery – Scratchers – picking the winning tickets using math

I’ve always been fascinated by how the lottery works. From a mathematical point of view spending money on lottery tickets is a complete waste of time and money. There are a few exceptions – when a lottery is poorly designed, it is possible to game the system and actually earn money – here’s a famous example of how MIT Students Won $8 Million in the Massachusetts Lottery.

The North American lottery system is a $70 billion-a-year business, an industry bigger than movie tickets, music, and porn combined. So general public seems to ignore the probability theory and continues to spend money in pursuite of the elusive multimillion grand prizes of the numerous lotteries.

I decided to take a closer look at Califortia State Lottery and came across a very nice analysis done by the Wizard of Odds. I do wholeheartedly agree with the advice given by the Wizard: “Don’t play in the first place. Every state lottery offers terrible odds. With few exceptions, it is the worst bet you can make.”

But I was intrigued by the following passage in the article: “The California Lottery is nice enough to indicate how many tickets for each win have already been cashed. If there is a game that is almost sold out, as evidenced by the small wins, with a high ratio of large wins still unclaimed, then it may mean the remaining unsold tickets are rich in big winners. The same principle as card counting in blackjack.”

I decided to take a close look at the odds of winning and the average return for each of the scratch card games CA Lottery offers and see if there are substantial changes in probability of winning and return on investment with the additional data provided by state lottery.

Here is a summary of my findings:

  1. More expensive scratchers have higher odds of winning and higher return:
    Scratch Card Average Returns
    Bet Average Return
    $1 52.49%
    $2 56.06%
    $3 57.57%
    $5 62.37%
    $10 67.75%
    $20 69.02%
    $30 72.63%
  2. There is indeed a potential advantage play in scratch card games, as with the information about the number of claimed prizes that CA Lottery publishes it is possible to calculate updated winning odds and returns – see the spreadsheet for details.
  3. Over time changes in the odds of winning and expected return can go up or down ~20%, so if you gamble, it definitely makes sense to run the numbers first
  4. As of February 10, 2018 Set For Life scratchers have the highest estimated return of 75.65%
  5.  As of February 10, 2018 $10 Million Dazzler scratchers have the highest estimated chance of winning of 36.5%
  6. In the unlikely event you win a grand prize – be aware of the fine print – the lottery is not going to pay it to you straight away. You’ll have a chance of taking home ~1/2 of it before taxes immediately or getting it paid out as installments over 25 years or so.

My full analysis is in this spreadsheet – it automatically downloads the latest stats on claimed prizes and winning odds from, so results you see there might differ from the examples I provided above.

And as a closing note – my general advice is “do not gamble”. If you do, then California Lottery is a good choice – its mission is to maximize supplemental funding for the state’s public schools, which is very similar to what we have as a mission for Veikkaus – the Finnish national betting agency. At least the money you lose will go to a good cause.