Phỏm (Tá Lả) Probabilities

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Phỏm (Tá Lả) Probabilities — Odds & Statistics | Card Games Academy

Starting Player (10 cards): C(52,10) = 15,820,024,220
Other Players (9 cards from remaining 42): C(42,9) = 1,098,847,070

Hand Quality	Probability	Recommended Play
1 complete Phom dealt (3+ consecutive or set)	22%	Excellent start
2 potential Phom sets (Ca)	34%	Strong starting hand
3+ connected cards (potential Phom)	68%	Play aggressively
No connected cards	8%	Defensive play required

Value Category	Expected Count	Probability of 0	Probability of 3+
Low cards (A-5)	2.8	3.2%	38%
Mid cards (6-9)	2.8	3.2%	36%
High cards (10-K)	3.5	0.8%	52%

Starting Cards	Probability of Completing	Average Turns Needed
2 of a kind (pair)	22%	4-5 turns
1 card + need 2 more	2.1%	Rare completion
3 of a kind (complete)	1.5% (dealt)	Instant

Starting Cards	Probability of Completing	Average Turns Needed
2 consecutive same suit	31%	3-4 turns
2 cards with 1 gap same suit	18%	5-6 turns
1 card + need 2 connected	8.5%	Unlikely
3+ consecutive same suit	12% (dealt)	Instant

Round	P(Having >= 1 Phom)	P(Having >= 2 Phom)	P(Having 3 Phom/U)
Round 1 (after deal)	22%	1.5%	0.1%
Round 3	52%	8%	1.2%
Round 5	76%	22%	4.5%
Round 7	91%	42%	8.8%
Round 9 (final)	97%	61%	12%

Scenario	Probability	Notes
U on first discard pick-up	~0.8%	Extremely rare
U by round 5	~4%	Needs strong starting hand
U by round 9 (final)	~12%	Average for any single hand
Any player achieving U in a 4-player game	~38%	At least one player per game

Factor	Impact on U Probability	Description
Starting with 2 potential melds	+15%	Strong foundation
Starting with 1 complete meld	+20%	Excellent start
Opponents feeding you cards	+12%	Lucky discards
Starting with no connected cards	-8%	Must rebuild entirely

Situation	Probability per Draw
Need 1 specific rank (3 copies remaining)	~6.5%
Need 1 specific rank (2 copies remaining)	~4.3%
Need 1 of 2 ranks (6 copies remaining)	~13%
Need 1 of 2 ranks (4 copies remaining)	~8.7%
Need 1 card to complete a sequence	~8.7%
Any card that extends existing meld	~17-22%
Drawing any card you can eat	~25-30%

Remaining Cards in Deck	Draw Probability (1 of 8 needed)	Draw Probability (1 of 4 needed)
30 cards	26.7%	13.3%
20 cards	40%	20%
10 cards	80%	40%
5 cards	100%+ (overlap possible)	80%

Your Discard	Risk Opponent Can Use It	Strategy
Low card (3-5)	Low (~15%)	Safe discard
Mid card (6-9)	Medium (~25%)	Moderate risk
Face card (10-K)	High (~35%)	High risk
Card matching known pair in play	Very High (~55%)	Avoid if possible
Card adjacent to known sequences	Very High (~60%)	Extremely dangerous

Scenario	Probability	Risk Level
You discard on final turn, opponent U	~2.8%	High penalty
You discard mid-game, opponent eats 3	~8.5%	Moderate
Safe discard (3+ copies already shown)	~0.5%	Very low

Scenario	Probability	Notes
Starting with no potential melds	8%	High risk of Mom
Mom by final round (bad starting hand)	3%	Uncommon but devastating
Any player getting Mom in 4-player	~11%	Roughly 1 in 9 games

Action Taken	Mom Probability Reduction
Aggressive eating in early rounds	-40% relative
Prioritizing run formation over sets	-25% relative
Sending cards to opponents' melds	-15% relative

Option A: Eat the 7H (form a set of three 7s)
- You complete a Phom immediately
- You reveal partial hand information
- EV: +3 points reduced from deadwood

Option B: Draw from deck instead
- You keep hand information hidden
- Probability of drawing something useful: ~20%
- Risk: The 7H might not come around again

Discard 5C:
- Risk opponent uses it: ~15%
- If used, penalty: 1 bet

Discard QS:
- Risk opponent uses it: ~35%
- If used, penalty: 1 bet + possible meld for opponent

EV(5C) = 0.15 x (-1 bet) = -0.15 bets
EV(QS) = 0.35 x (-1 bet) = -0.35 bets

Event	Probability (4 players)	Impact
At least one player gets U	~38%	Game ends immediately
At least one player gets Mom	~11%	Penalty doubled
Den Lang triggered	~5.5%	Severe penalty
Game goes to full 9 rounds	~62%	Standard ending

Information Available	Probability Insight
3 of a rank seen	Fourth is in someone's hand
No face cards discarded yet	Opponents holding high cards
Opponent eats twice	They are close to U - be cautious
Opponent never eats	They may have strong hidden melds

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