Poker Analytics & Probability

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What this quiz asks

1. In a standard 52-card deck, what is the exact number of possible unique two-card starting hands (combinations) a player can be dealt?
2. If you hold a pocket pair (e.g., AA) in Texas Hold'em, how many distinct ways can the remaining 50 cards in the deck form the other two cards of the same rank to complete a set (three-of-a-kind) on the flop?
3. Calculate the number of ways to form a flush (all five cards of the same suit) using exactly 5 cards from a 52-card deck, excluding straight flushes.
4. In a game of Texas Hold'em, you hold 7-8 suited. How many unique 'flop' combinations (3 cards) contain at least one card of your suit, given that two cards of your suit are already in your hand?
5. In a $100 pot, your opponent bets $50. You need to call $50 to see the showdown. What is the required equity you need to justify a call based on pot odds?
6. You have a flush draw with 9 outs on the turn. With one card to come, what is your approximate equity percentage?
7. If the pot is $400 and your opponent bets $200, what is the minimum equity required to make a profitable call?
8. You hold a draw with 25% equity. The pot is $300 and the bet to you is $150. Is this a profitable call?
9. You are facing a bet of $50 into a $100 pot. Your equity in the hand is 25%. Calculate the Expected Value (EV) of calling the $50 bet.
10. A pot contains $200. An opponent bets $100. You need to call $100. If your win probability is 30%, what is the EV of your call?
11. You hold a draw with 9 outs on the turn (44 cards remaining). The pot is $400. Villain bets $200. What is the EV of calling, assuming you fold if you miss on the river?
12. You are closing the action. Pot is $500, bet is $200. You need 28.5% equity to break even. If your actual equity is 30%, what is your total profit/loss expectation in dollars?
13. A professional poker player with a bankroll of $10,000 plays a game where the standard deviation of their hourly results is $50. After 100 hours of play, what is the standard deviation of their total profit?
14. If a player has a 2% risk of ruin for a $20,000 bankroll, what is the approximate probability of ruin if they increase their bankroll to $40,000 while keeping the same betting structure and win rate?
15. A player has an hourly win rate of $20 and a standard deviation of $100 per hour. After 400 hours, what is the probability that the player is showing a net loss?
16. If a player's bankroll requires 500 big blinds (BB) to survive a 5% risk of ruin, and they decide to play in a game with 2x the variance, what is the new required bankroll (in BB) to maintain the same 5% risk?
17. In a heads-up poker scenario, you assign a 60% prior probability to your opponent holding a flush draw. After they bet into a board that contains no flush possibilities, you update your belief using Bayesian Inference. If the likelihood of them betting given they have a flush draw is 0.2, and the likelihood of them betting without a flush draw is 0.8, what is the posterior probability they have the flush draw?
18. You are modeling a player's range. Initially, you believe they have a set 10% of the time. They check-raise a dry board. Your model suggests that if they have a set, they check-raise 90% of the time, but if they have air, they check-raise only 5% of the time. What is the updated probability they have a set?
19. Consider a situation where you believe an opponent bluffs with frequency P(Bluff) = 0.3. If they bet into you, and you know P(Bet|Bluff) = 0.8 and P(Bet|Value) = 0.4, determine the probability they are bluffing after they bet. If they then bet again on the turn, and you assume P(Bet2|Bluff) = 0.6 and P(Bet2|Value) = 0.9, what is the final posterior probability they are bluffing?
20. An opponent's range consists of 40 combos of value hands and 60 combos of bluffs. They bet 100% of their value hands and 50% of their bluffs. If they bet, what is the probability they have a bluff? Furthermore, if they then check-raise 20% of their remaining bluffs and 80% of their value hands, how does the posterior change?
21. In a simplified heads-up GTO model, you face a pot of $100 and a bet of $100. To prevent your opponent from exploiting you with a bluff, what is the minimum frequency of calls (defensive range) required to make your opponent indifferent to bluffing?
22. Given a pot of $200 and a bet of $100, what is the GTO bluffing frequency (the percentage of hands you should check-raise or bet as bluffs) to remain unexploitable?
23. In a GTO scenario, you have a range consisting of 40 value hands and 20 bluffs. If you bet $100 into a $100 pot, what is the required ratio of value to bluffs for your range to be theoretically balanced?
24. You are playing a game where you have a 30% equity in a $500 pot. If you employ a GTO strategy that requires a 25% bluff frequency, what is the absolute minimum EV of your bluffing component alone if the bet size is $200?

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