Scratchie Odds, Worked Through Ticket by Ticket

Instant scratch tickets from China Sports Lottery, 2018; these are Chinese tickets, not Australian scratchies. Photo: Yohannshen, CC BY-SA 4.0

What the headline odds actually count

The Lott, which sells Instant Scratch-Its, says in its help centre article on game odds that the odds of winning any prize on an Instant Scratch-Its game are about 1 in 4, and that the top-prize odds for a specific game are printed on the back of the ticket. That figure answers one question only: how often a ticket shows some prize. It does not say how big the prizes are, or how many of them simply return your money.

To answer those questions you need the whole prize table: every prize level and the number of tickets carrying it. We could not open the individual game pages on thelott.com, so for the worked examples below we used Lotterywest, the Western Australian government lottery, which publishes complete tables for its Scratch'n'Win tickets. The method is the same for any instant ticket with a published table.

Lotterywest attaches three conditions to its figures, and they matter for the maths. Overall odds refer to all tickets distributed in WA for the entire game, not to a single batch or roll. Winning and losing tickets are randomly distributed. And every prize, including the top prize, is only available until it is won and claimed.

Worked example: a $5 ticket

The Cue the Prize ticket (game 773) costs $5. Lotterywest lists 750,000 tickets in the game and overall odds of 1 in 4.16.

Cue the Prize prize table as published by Lotterywest, with our prize-pool column.

Step 1: count the winners

Adding the second column gives 180,175 winning tickets. Then 750,000 ÷ 180,175 = 4.163, which matches the published 1 in 4.16.

Step 2: total the prize pool

Adding the last column gives $2,400,000 in prizes. Selling all 750,000 tickets at $5 brings in $3,750,000. So the game hands back $2,400,000 ÷ $3,750,000 = 64.0% of what players spend.

Step 3: expected value per ticket

Expected value is the prize pool spread across every ticket: $2,400,000 ÷ 750,000 = $3.20. On average, each $5 ticket returns $3.20, an expected loss of $1.80, or 36 cents in every dollar.

Step 4: strip out the prizes that only return your stake

Of the 180,175 wins, 66,000 are $5 prizes, the same as the ticket price. That is 66,000 ÷ 180,175 = 36.6% of all wins. A ticket that pays more than it cost is one of the remaining 114,175, and 750,000 ÷ 114,175 = 6.57. The real chance of coming out ahead on one ticket is about 1 in 6.6, not 1 in 4.

Step 5: how much is the top prize worth to you?

The single $100,000 prize adds $100,000 ÷ 750,000 = 13.3 cents to the expected value. Without it, the remaining $2,300,000 spread over 750,000 tickets is $3.07. Because the top prize is only available until it is won and claimed, a ticket bought after it has gone is worth that lower figure on average.

Step 6: a small run of tickets

The chance that one ticket wins nothing is 569,825 ÷ 750,000 = 75.98%. As a rough approximation that treats each ticket as independent (strictly, they are drawn from a fixed pool), the chance that four tickets all lose is 0.7598 × 0.7598 × 0.7598 × 0.7598 = 0.333. About one buyer in three who picks up four tickets will scratch nothing at all.

Comparing price points

We ran the same steps on two more Lotterywest tickets: Wild 1s (game 35) at $1, with 1,500,000 tickets, and Break the Bank (game 981) at $20, with 600,000 tickets.

Three Lotterywest tickets compared, our calculations from the published prize tables.

The working for the other two columns follows the same pattern. Wild 1s: 342,659 winning tickets; 142,000 of them are $1 prizes, and 142,000 ÷ 342,659 = 41.4%; the other 200,659 give 1,500,000 ÷ 200,659 = 7.48. Break the Bank: 174,563 winning tickets and $8,190,000 ÷ 600,000 = $13.65. Its two $250,000 top prizes make up $500,000 of the pool, so without them the expected value drops to $7,690,000 ÷ 600,000 = $12.82.

What the numbers say

• Every ticket loses money on average. Across these three games the return ranges from 57.5% to 68.25%, so the expected loss is between 31.75 and 42.5 cents in every dollar.
• In these three games, a dearer ticket returned a larger share. That is a pattern in three tables, not a rule, and a bigger share of a bigger stake is still a bigger expected loss in dollars: $6.35 on the $20 ticket against 42.5 cents on the $1 ticket.
• The headline odds overstate profit. On the two cheaper tickets, more than a third of the wins only hand back the purchase price.
• The top prize is a small slice of the value. On Cue the Prize it is worth 13.3 cents per ticket.

These returns are low by casino standards; our look at which table games pay back more sets the comparison out.

The same expected-value method works for any game that publishes its prize structure. Keno is the obvious next candidate, since it is another low-stake game sold through the same outlets; our notes on keno cover how that game plays.

Online scratchies are a different matter

Buying a ticket from an authorised Australian lottery is legal for adults 18 and over. Instant scratch games offered by offshore websites are not: the ACMA lists scratchies among the illegal online services that operators cannot offer Australians, and warns that players have little chance of recovering money from those sites.

Treat the expected loss as the price of the entertainment and set the amount before you buy, not after a near miss. If scratchies stop being a small treat, the National Gambling Helpline on 1800 858 858 is open 24 hours a day, and our responsible gambling page lists other support.