How to Think About What a Tie or Push Actually Does to Your Expected Loss

Most gambling explanations treat a bet as having two possible endings: you win or you lose. Many real games, however, include a third possibility. A hand of baccarat can push on certain totals. A blackjack hand can tie. A sports bet can land exactly on the spread and be graded as no action. Understanding what that third outcome actually does to your math is worth a few minutes of careful thought.

The Basic Setup: A Two-Outcome Bet

Start with a simple hypothetical. Imagine a coin-flip game where the house pays you $0.95 for every $1.00 you risk, and you lose your $1.00 when the coin lands the wrong way. Each outcome has a 50% probability.

Expected value per $1.00 bet:

• Win probability x net gain: 0.50 x $0.95 = +$0.475
• Loss probability x net loss: 0.50 x (-$1.00) = -$0.500
• Expected value = +$0.475 - $0.500 = -$0.025

You expect to lose $0.025, or 2.5 cents, per $1.00 wagered on average over many bets. That 2.5% is the house edge in this example.

Adding a Push: What Changes and What Does Not

Now suppose the game is modified. A third outcome is introduced: a tie, which returns your $1.00 stake and settles for zero net gain or loss. Assign these hypothetical probabilities:

• Win: 48%
• Loss: 48%
• Tie: 4%

The win and loss probabilities each dropped by 2 percentage points to make room for the tie. The payout on a win is still $0.95 per $1.00 risked.

New expected value per $1.00 bet:

• Win: 0.48 x $0.95 = +$0.456
• Loss: 0.48 x (-$1.00) = -$0.480
• Tie: 0.04 x $0.00 = $0.000
• Expected value = +$0.456 - $0.480 = -$0.024

The expected loss per bet fell slightly, from 2.5 cents to 2.4 cents. That looks like a mild improvement for the player.

But here is the part worth focusing on: the expected loss is now spread across fewer resolved bets. When a tie occurs, no money changes hands, but a round of the game is consumed. Your time and attention were spent, yet neither side of the ledger moved.

Why Ties Affect Your Cost Per Hour, Not Just Per Bet

Imagine you play 100 rounds in a session. In the two-outcome version, all 100 rounds produce a win or a loss. In the three-outcome version with 4% ties, roughly 4 of those rounds return your stake and 96 produce a real result.

Your per-round expected cost across all 100 rounds in the three-outcome example:

• 100 rounds x (-$0.024) = -$2.40 expected

In the two-outcome version:

• 100 rounds x (-$0.025) = -$2.50 expected

So introducing a tie with those hypothetical numbers saved you about $0.10 in expected losses across a 100-round session. That is a real but small improvement.

However, if ties are resolved instantly and the game just keeps moving at the same pace, you are still spending the same amount of time at the table. Ties in that scenario are nearly neutral for your hourly cost because the game speed absorbs them without slowing down.

If instead a tie genuinely pauses the game and you get that round back at zero cost, then ties are a mild structural benefit: you are effectively getting a free round.

A Common Misreading: Ties as Miniature Wins

Some players intuitively treat a push as a small victory. Getting their money back after expecting to lose can feel like good news. Mathematically, a tie contributes zero to your expected value calculation. It is neither a gain nor a loss. That feeling of relief is understandable, but it should not change how you assess the game's true cost.

A related mistake is assuming that a game with frequent ties is therefore player-friendly. Whether ties help or hurt you depends entirely on how the three probabilities are distributed and what the payout is on a win. A game could in principle shrink the win probability significantly to fund a high tie rate, which would make the player worse off even though the push itself costs nothing.

How to Calculate This for Any Three-Outcome Game

The expected value formula extends cleanly to any number of outcomes. For a three-outcome bet:

EV = (P_win x Net_win) + (P_loss x Net_loss) + (P_tie x 0)

Because the tie term is always zero, it drops out of the calculation entirely. What matters for expected value is only the win and loss probabilities and their associated payouts. The tie probability matters indirectly because it forces P_win and P_loss to sum to less than 1, which changes both of those terms.

To apply this in practice, you need three pieces of information:

1. The probability of winning the bet
2. The net payout if you win
3. The probability of losing the bet

You do not need the tie probability explicitly because it is simply 1 minus the other two. Once you have the win and loss probabilities and their payouts, the formula above gives you the expected value per unit wagered.

Assumptions and Limitations

The examples above use hypothetical numbers constructed to illustrate the mechanics. Real games have their own specific probabilities, which depend on rules that vary by casino, table, and version of the game. Published house edge figures for games with ties already account for the push probability in their calculations, so you do not need to adjust those figures yourself if you are working from a reliable source.

Expected value describes an average across a large number of trials. In any single session, your result can differ significantly from the expected value in either direction, regardless of whether ties are present. The presence of ties does not reduce that short-run variability in any simple way.

If you want to compare two versions of a game where one has a tie outcome and the other does not, the most honest comparison is to calculate the expected loss per round for each version using the actual probabilities and payouts for that specific game, then consider how game speed affects your hourly exposure.

The Practical Summary

A tie outcome in a gambling game is mathematically neutral in itself. It contributes zero to your expected gain or loss. Its real effect is indirect: it absorbs some probability from the win and loss columns, which changes your per-round expected cost. Whether that change benefits you depends on how the probabilities shift, not on the existence of the tie alone. Keeping those two things separate, the mechanics of the push and the redistribution of probabilities, gives you a clearer picture of what any three-outcome game actually costs.