How Is Casino War Tie Probability Calculated? Eight Decks, Ties, and First-Deal Odds

Direct answer

LINBEI War uses an eight-deck, 416-card finite shoe. After the first card is known, 31 of the remaining 415 cards share its rank, so first-deal Tie=31/415.

Quick facts

Shoe

8 decks · 416 cards

First-deal Tie

31/415≈7.4699%

Player win

192/415≈46.2651%

Dealer win

192/415≈46.2651%

01

Why Tie is 31/415

Across eight decks, every rank begins with 32 cards. Once the first card has been drawn, 31 cards of that same rank remain among 415 total cards.

The second card therefore matches rank with probability 31/415≈7.4699%. This is finite sampling without replacement, not exactly 1/13.

02

Player-win and dealer-win probability

After excluding ties, rank ordering is symmetric between Player and Dealer, so their first-deal win probabilities are equal.

P(player win)=P(dealer win)=(1−31/415)/2=192/415≈46.2651%. The three first-deal outcomes sum to 100%.

03

General D-deck formula

With D complete 52-card decks, each rank appears 4D times. After the first draw, 4D−1 same-rank cards remain out of 52D−1 total.

Therefore P(tie)=(4D−1)/(52D−1), and the remaining non-tie probability is split equally between Player and Dealer.

04

EV after a Tie depends on strategy and paytable

The first-deal tie probability is exact, but expected value after a tie depends on Surrender versus Go To War, extra exposure, War rules, repeat ties and payout terms.

LINBEI Practice Tie / War payouts are UX-validation rules. This guide does not claim they match another operator's production paytable or turn first-deal probability into a full game EV.

EXAMPLE

Eight-deck first deal

• Shoe=8×52=416 cards
• Same-rank cards after first draw=8×4−1=31
• P(Tie)=31/415≈7.4698795%
• P(Player Win)=P(Dealer Win)=192/415≈46.2650602%

Frequently asked questions