The math behind New York LOTTO

New York LOTTO is a simple lottery jackpot game offered by the New York Lottery, with variable prizes in all tiers, and consisting of two draws per week. This article covers the statistical indicators of the NewYork LOTTO and what distinguishes this game from other lottery games in its category.

Category

New York LOTTO is a combinatorial lottery game drawing from one pool: seven numbers drawn from 1 to 59, forming a 6-number base draw and a bonus number.

• Size of the draw: 7, that is, 6 + 1.
• Size of a played line: 6, with no separate number as the bonus number.
• Cost of a line: $0.5, with two lines grouped on one ticket of $1.

The game is advertised as having multi-draw features, allowing you to play the same ticket for several successive draws (up to 26).

Prize schedule

There are five prize tiers, defined as matching 3 to 6 numbers in the draw.

Unlike similar games with a bonus ball, such as Powerball, matching the bonus ball is only included in the definition of one tier, namely the second. Hence, the top prize (jackpot) is not defined as matching the base draw and the bonus ball, but six numbers in the base draw.

All the prizes are variable, as a share of the prize fund, which is up to 40% of the sales, as follows:

There is an abrupt jump from the first tier to the second tier, and the shares allocated to tiers second to the fifth range within a narrow interval between 5.25% and 6%.

To estimate the average amount of each prize, we use the average jackpot value ($7.3 million over the last 12 months) and the expected number of winners for each prize tier.

The resulting average prizes closely align with the actual payout scheme for most lotteries. Hence, although the percentage schedule looks unusual and does not preserve the inverse order of the probabilities of winning, when the average number of winners is factored in, the prize schedule comes to preserve that order, with two abrupt jumps – from the first to the second tier and from the second to the third tier.

Odds of winning

The game advertises the odds of winning based on one line. The odds of winning with one ticket (holding two lines as two independent plays) are roughly twice as high as those odds.

The overall odds of winning are about 1 in 46 per line and roughly 1 in 23 for one ticket.

The magnitude of these odds aligns with those of the most combinational lotteries based on a 6-number draw. The highest odds (for the fifth tier) apparently indicate a potential for a volume purchase strategy; the average prize in this tier (with the number of expected winners factored in) is a barrier to crediting such a strategy. Even for higher jackpots, the argument stands. For instance, at a $70 million jackpot, the expected prize in the fifth tier is about $9. Still, the volume strategy remains an option, based on expectation in the long run, for higher jackpots, also targeting higher prizes.

Average win and fairness

The next table notes the average win for each tier, computed using the average prize amounts based on a $7.3 million average jackpot.

This means, for example, that over the long run, a win in the fourth tier is distributed as $0.0135 average win per line played (assuming the prize is won with the relative frequency of about 1 in 2,180 draws).

The next table lists the fairness of each prize relative to its probability, computed under the same assumption regarding the average jackpot:

Such figures indicate a poor fairness odds-prize. Even Powerball has greater fairness (over 70% for the average rolling-over jackpot). The numbers can yet change for higher jackpots.

Expected value and house edge

Assuming 40% as the share of the ticket sales that goes to the prize fund, the expected value of the New York LOTTO play is –40% (the return from a $1 ticket is $0.40), or –20% for one line. This gives the game a 40% house edge for one ticket or 20% for one line, less than the typical house edge of 50%-60% of lotteries in similar categories.

According to the given statistical indicators, the game may become positive-EV for a certain jackpot value. The theoretical breakeven point of the positive expectation is $20.3 million.

In reality, the break-even value is higher because multiple winners split the jackpot, taxes apply after a win, and the cash value is lower than the annuity jackpot.

Accounting for these, the practical positive-EV region is closer to $35 million – $50 million.

Final remarks

The multi-draw features of the New York LOTTO have no statistical relevance, as playing the same ticket for several successive draws does not change the statistical indicators of the game, nor does it increase your chances to win in a given play.

The low cost of a line makes the game suitable for being exploited mathematically, not necessarily for targeting a certain prize in a single or short- to medium-run play, but for an optimal strategy in the long run based on passing the break-even value of the jackpot.

This assumes a large number of tickets bought in a team, spreading numbers to avoid overlapping, and playing only when the jackpots exceed the break-even value (over $35 millions). Such a strategy does not guarantee winning, but maximization of profit and minimization of loss, interpreted as: If sufficient wins occur in the long run, you may expect the overall win to exceed the previous loss.