Bridge Hand Probability Puzzles: Famous Problems Every Player Should Know

Bridge is full of probabilities that feel wrong. Suits split unevenly more often than evenly. Finesses fail at the worst times. And sometimes, the mathematically correct play looks completely backward.

These puzzles aren’t just brain teasers. They teach you how to think about probability at the table. Master these, and you’ll stop making gut-feel errors that cost you tricks.

Puzzle 1: The Monty Hall Problem at the Bridge Table

You’re declaring 6♠ and need to locate the ♥K. Dummy has AQ in hearts, you have small cards.

You lead a heart toward dummy. LHO thinks for a while, then plays low. You finesse the queen…and it wins.

Later, you’re back in hand. You lead another heart toward dummy. LHO plays low again.

Should you:

A) Finesse the queen again (playing RHO for the king)
B) Play the ace (playing LHO for the king)

Most players say B. “LHO thought about it the first time. They must have the king.”

Wrong.

The answer is A. Finesse again.

Why?

This is the bridge equivalent of the Monty Hall problem. LHO’s “choice” to play low is information—but not the information you think.

When LHO had Kx, they had to play low (playing the king would give you 2 tricks). When LHO had xx, they played low because that’s all they could do.

The fact that LHO thought about it suggests they had a decision. If they had xx, there’s no decision—you just play low. The hesitation suggests Kx.

But here’s the twist: even if they had the king, they have to play low. So the hesitation doesn’t actually tell you they have the king. It tells you they might have the king, or they might be thinking about count signals, or they might be messing with you.

The correct play is still to finesse. Because:

  • If RHO had Kx originally, they still have the king now.
  • If LHO had Kx originally, the finesse loses…but you were always losing a heart trick anyway.

The math: Before the first finesse, the king was 50-50. The first finesse won, which means RHO didn’t play the king. That doesn’t mean RHO doesn’t have the king—it means either RHO has the king and didn’t play it (because they didn’t have a singleton), or LHO has it.

Actually, wait. Let’s think about this more carefully. You’re missing the ♥K and some small hearts.

If LHO had K alone, they’d have to play it. So LHO either had Kx+ or nothing relevant.

The real Monty Hall: This puzzle is actually about Restricted Choice (covered in another article). When LHO could have played the king but didn’t, you update your probabilities. Finessing again is correct.

The Lesson

When an opponent makes a “choice,” that choice is information. Use it.

Puzzle 2: The Birthday Paradox in Suit Splits

You’re missing 13 cards in a suit (opponents have all 13). What are the odds that the suit splits exactly 7-6?

A) About 50%
B) About 30%
C) About 10%

Most players guess A or B. After all, 7-6 is close to even.

Answer: C. About 10%.

Why?

There are lots of ways to split 13 cards. Exactly 7-6 is just one of them. You could have 8-5, 9-4, 10-3, etc.

The most common split for 13 outstanding cards is actually 7-6 or 8-5, each happening about 10-15% of the time. But any specific split is unlikely.

This is like the birthday paradox: in a room of 23 people, the odds that two people share a birthday are over 50%. But the odds that you specifically share a birthday with someone else are much lower.

Bridge distributions work the same way. Lots of possibilities, each individually unlikely.

The Lesson

Don’t assume “balanced” distributions are common. With many outstanding cards, all distributions are uncommon.

Puzzle 3: The Two-Headed Queen

Dummy has ♥ AJ10. You have ♥ xxx. You’re missing the KQ.

You lead toward dummy, RHO plays the queen. You win the ace. Both opponents follow small on the next round.

You’re back in hand. You lead toward dummy, RHO plays low.

Should you finesse the 10 (playing RHO for Q alone) or play the jack (playing RHO for KQ)?

Before you answer, here’s the twist: RHO is a tricky player who, when holding KQ, always plays the queen first (never the king).

Now what?

A) Finesse the 10
B) Play the jack
C) Flip a coin

Answer: B. Play the jack.

If RHO always plays the queen from KQ, then seeing the queen doesn’t give you any information. RHO played the queen either because:

  • They had Q alone (forced to play it), or
  • They had KQ (and always play the queen first).

Both are equally likely. You’re back to 50-50.

But wait—Restricted Choice says you should finesse the 10, because RHO could have played the king from KQ.

The catch: RHO never plays the king from KQ. So Restricted Choice doesn’t apply.

When opponents don’t randomize their equals, you lose the Restricted Choice edge.

The Lesson

Restricted Choice works because opponents play equals randomly. Against players with known habits, adjust your strategy.

Puzzle 4: The Singleton King

You’re in 7NT. You have AQ1098 in spades opposite xxx. You’re missing the KJ.

You lead toward dummy, RHO plays the king.

Should you:

A) Win the ace and finesse the 10 on the way back
B) Win the ace and play for the drop
C) Duck, hoping RHO has a singleton king

Answer: C. Duck (usually).

This is a weird one. If RHO has a singleton king, ducking costs you nothing—you were always getting 4 tricks. But if RHO has Kx or KJx, ducking might gain.

Wait, that doesn’t sound right. Let’s think through the cases:

  • RHO has K alone: Duck wins 5 tricks. Win the ace and you win 4 tricks.
  • RHO has Kx: Duck, LHO wins the jack, you get 4 tricks. Win the ace, finesse the 10, you get 4 or 5 tricks depending on who has the jack.
  • RHO has KJ (doubleton): Duck, LHO wins, you get 3 tricks. Win the ace, finesse the 10, you get 5 tricks.

Actually, this is a safety play decision. If you’re in 7NT and need all 5 tricks, you must finesse (hoping RHO has K alone or KJ). If you only need 4 tricks, ducking is safer.

The Lesson

Think about what you need, not just what’s possible. Sometimes the “safe” play is better than the “greedy” play.

Puzzle 5: The Missing Spot Card

You have AKQ1098 in a suit. You’re missing the J7654.

You cash the ace, both follow (LHO plays the 4, RHO plays the 5). You cash the king, both follow (LHO plays the 6, RHO plays the 7).

Is the jack more likely with LHO or RHO?

A) LHO (they played lower cards first)
B) RHO (they played higher cards first)
C) Equal

Answer: C. Equal (unless you know their carding agreements).

The spot cards LHO and RHO played don’t tell you where the jack is—they tell you about count. If LHO played low-high (4 then 6), they might be showing an even number. If RHO played high-low (7 then 5), they might be showing an even number.

But different players use different carding agreements. Some play upside-down count. Some give attitude, not count.

The jack is still 50-50 based on this information alone.

The Lesson

Spot cards signal count or attitude, not location of honors. Don’t confuse the two.

Puzzle 6: The Uninformative Bid

LHO opens 1♠. You’re declaring 3NT and need to locate the ♦K.

Is the king more likely with LHO or RHO?

A) LHO (they opened, so they have it)
B) RHO (LHO already has spade length, so less room)
C) Still 50-50 until you know more

Answer: C. Still 50-50 (mostly).

Opening 1♠ shows 12+ points and 5+ spades. The ♦K is 3 HCP. LHO could have 12 HCP with the king, or 15 HCP with the king, or 14 HCP without the king.

You need more information. If LHO shows up with 10 HCP in other suits (say, the ♠AK), and they opened 1♠ (showing 12+ HCP), they probably have 2-3 more HCP somewhere. The ♦K is a good candidate.

But without seeing more cards, the opening bid alone doesn’t tell you much.

The Lesson

Opening bids give ranges, not specifics. You need to combine bidding with play to narrow the range.

Puzzle 7: The Falsecard

You’re declaring 4♥. Dummy has ♠ AJ10, you have ♠ xxx. You lead toward dummy, RHO plays the king.

You win the ace. Later, you lead toward dummy again. RHO plays low.

Should you finesse the 10 or play the jack?

Answer: Finesse the 10 (Restricted Choice).

But here’s the puzzle: What if RHO is a great player who knows about Restricted Choice?

If RHO has KQ, they might play the king to make you think they have a singleton king. Now when you finesse the 10, it loses to the queen.

Should you adjust?

Answer: No. Still finesse the 10.

Here’s why: If RHO falsecards from KQ by playing the king (hoping you’ll finesse the 10), then you should finesse the 10—because that’s what they’re expecting. But if they have K alone, the finesse wins.

Wait, that doesn’t make sense. Let me think through this again.

If RHO has K alone, the finesse wins. If RHO has KQ and falsecards the king, the finesse loses. So RHO’s falsecard works.

But here’s the catch: falsecarding from KQ by playing the king is only good if you know declarer uses Restricted Choice. Against weak players, it’s a wasted card.

Against good players, RHO will falsecard sometimes (to keep you honest) but not always. So you still have an edge by finessing.

The Lesson

Don’t overthink. Play the percentages. Even against good opponents, math beats guesswork.

Puzzle 8: The 13-0 Break

You’re in 6♠ with a solid 8-card trump suit (AKQ10xxx opposite xx). You’re missing 5 trumps, including the jack.

On the first round, LHO shows out. RHO has all 5.

What are the odds of this happening?

A) 1 in 20
B) 1 in 50
C) 1 in 200

Answer: A bit worse than 1 in 20 (about 4%).

Missing 5 cards, a 5-0 break happens about 4% of the time. It’s rare, but not that rare.

You’ll see it every few sessions.

The Lesson

Even unlikely distributions happen. Be ready for them. When you have 8+ cards in a suit, cash the top honors from the right hand to pick up 5-0 breaks.

Puzzle 9: The Principle of Symmetry

Dummy has ♦ AJ10, you have ♦ xxx. You’re missing the KQ.

You can lead from either hand. Should you lead from hand toward the AJ10, or from dummy toward your xxx?

A) From hand (toward the tenace)
B) From dummy (toward nothing)
C) Doesn’t matter

Answer: A. From hand toward the tenace.

Leading toward the tenace gives you the finesse. Leading from the tenace gives you…nothing.

But here’s the puzzle: What if you’re in a suit contract and can ruff?

Now if you lead from dummy toward your xxx and RHO plays the king or queen, you ruff. If LHO has KQ, you ruff both honors. You get 3 tricks.

If you lead from hand toward the AJ10, you finesse normally. You get 2 tricks most of the time, 3 tricks if LHO has both honors.

The ruffing finesse is often better than the regular finesse.

The Lesson

In suit contracts, shortness is power. Use it.

Puzzle 10: The Self-Fulfilling Prophecy

You’re in 3NT. You have 8 top tricks. You can make your 9th trick by:

A) Finessing the ♥Q (50%)
B) Finding diamonds 3-3 (36%)

Which line do you choose?

Most players say A. 50% is better than 36%.

But wait. You can try both. Cash a few rounds of diamonds first. If they split 3-3, you’re home. If they don’t, fall back on the heart finesse.

Your combined chances: 36% (diamonds work) + 64% × 50% (diamonds fail, finesse works) = 36% + 32% = 68%.

That’s way better than either line alone.

The Lesson

Don’t commit to one line when you can try two. Test the percentages in the right order (lower percentage first, so you still have the finesse as backup).

The Deep Takeaway

Probability in bridge isn’t just about memorizing percentages. It’s about:

  1. Combining chances (try line A, then line B)
  2. Updating probabilities (Restricted Choice, vacant places)
  3. Thinking about what you need (safety plays vs greedy plays)
  4. Using opponent behavior (hesitations, falsecards)

These puzzles train your brain to think probabilistically. Not just “what’s more likely,” but “what should I do given what I know.”

That’s the skill. That’s the game.

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