Two-up odds: the maths of a fair ring and the casino house edge

The chances on each toss

Two coins can land four equally likely ways: head-head, head-tail, tail-head and tail-tail. Only one of the four is two heads, so heads come up one toss in four (½ × ½ = ¼). Two tails come up just as often. The two mixed results together make up half of all tosses, and in two-up they settle nothing.

In two-up, “odds” is the name of the mixed result, so when we talk about what a bet returns, we say it “pays” a price such as 25 to 1. Because half the tosses are odds, a bet takes two tosses on average to settle.

Why heads against tails is an even bet

Because odds are re-thrown, a bet is only ever settled by heads or tails, two results with the same chance. A side bet at even money with no commission has an expected result of zero for both players, which is why GambleAware NSW calls two-up a fair game. Its lesson adds: “Changing your bet each time does not affect the probability of winning.”

The diggers seem to have known their prices too. The War Memorial’s account of the 2/12 Commando records that “Other tail betters would bet 3/1 that heads would not be tossed twice”. Of the tosses that decide a bet, two heads in a row come up one time in four, so 3 to 1 is exactly the fair price.

The spinner’s run: three heads in a row

The spinner is playing a longer bet. Under the NSW rules of the ring, they can’t collect a dividend until the third head in a row (rule 11). Tails at any point cost them the centre (rule 12). Counting only the tosses that decide something, three heads in a row come up one run in eight (½ × ½ × ½ = ⅛).

If the winnings stay in the centre and ride on each toss, a $10 stake becomes $20, then $40, then $80. The spinner is $70 up one run in eight and $10 down the other seven. On average that is 1/8 × $70 minus 7/8 × $10, which is $8.75 minus $8.75: exactly zero. With no commission, the spinner’s run is as fair as a side bet.

What a commission costs

On Anzac Day, four jurisdictions write a no-cut rule into law. NSW, Queensland, South Australia and the ACT all bar commission on bets or winnings (NSW s 7, Qld s 179, SA s 19(2), ACT s 21). NSW and the ACT allow a fee or commission only when all of it goes to charity. Leave aside a charity collection, and the game stays at zero edge.

Western Australia is different. Permits under s 81 of the Gaming and Wagering Commission Act 1987, for venues over 100 km from Crown Perth, allow a “customary commission, not exceeding 10%”. The WA government’s two-up page applies it to the spinner’s winnings when they leave the centre, and to winning heads bets placed through the ringkeeper. If the 10% comes off the net win, here is what it costs.

For the spinner: 1/8 × ($70 − $7) − 7/8 × $10 = −$0.875 on a $10 stake, or 8.75%. For the heads bet: ½ × 0.9 − ½ = −0.05, or 5%. The permit system is explained on our page about the Kalgoorlie two-up shed. Broken Hill’s council may also take a commission under NSW s 10(4), and the Broken Hill game has its own page.

Casino two-up: the five-odds rule

Casino versions remove the even bet. In casinos, GambleAware NSW explains, you bet against the house only, and players can’t bet each other. Wizard of Odds describes The Star’s rules, and Wikipedia gives a table for Crown Perth. Both add one rule: if five odds come up in a row, every heads and tails bet loses, which both sources call odding out.

The maths takes one line. Five odds in a row happen (½)⁵ = 1 time in 32, or 2 times in 64. That leaves heads winning 31 rounds in 64, and tails the same. Under a five-odds rule, a heads bet at even money wins 31 times in 64 and loses 33, so the house keeps 2/64, or 3.125%.

Whether any Australian casino runs a table today is a separate question. Two-up is an authorised game at Crown Perth under the 1985 Burswood casino agreement. We couldn’t find a two-up table or rules on the casino’s site on 27 September 2026. Victoria’s regulator lists two-up among the approved games for Melbourne’s casino, and Crown Melbourne’s site has no two-up page.

Casino side bets and their edge

The same sources describe extra bets, and their prices vary by source. The fair price for five odds in a row is 31 to 1; the fair price for the spinner’s bet, three rounds won by heads, is about 7.8 to 1.

Over 32 rounds, a $1 five-odds bet at 28 to 1 wins $28 once and loses $1 31 times: a net loss of $3, or 3/32. At 25 to 1 the loss is $6, or 6/32. Wizard of Odds lists 30 to 1 in its table and 25 to 1 in its text; at 30 to 1 the edge is 3.125%.

The spinner’s bet wins only if heads take three rounds in a row, which happens (31/64)³ of the time, about 11.4%. Paying 15 to 2 on that leaves the house 3.40%, close to the edge on a plain heads bet.

What the numbers mean at the ring

No pattern of betting changes these numbers. Coins have no memory, so a run of heads says nothing about the next toss, and switching sides only swaps one even chance for another. In a fair ring, everyone expects to finish level over time, and the swings on the day are luck.

Two-up is still gambling, and even a fair game can empty your pocket in a short session. Decide your limit before the first toss and keep to cash you’re happy to spend. If it stops feeling like a bit of fun, Gambling Help Online is free and confidential on 1800 858 858, and our page on gambling responsibly lists other support.