You are playing Texas Hold’em against one other opponent.
Your two down cards are 6 of (diamonds) & 7 of (diamonds).
The first three cards to come up are 3 of (diamonds), 4♣ 9♠.
The next card to come up is 10 of (diamonds).
You and your opponent have both already bet $24,000 into the pot each. Your opponent now adds $10,000 to the pot. You only have $10,000 left in total.
While you consider what to do, your opponent, in an effort to rattle you, reveals his down cards. They are 10♣ 10♠.
The Problem
Should you call his bet by tossing the rest of your money into the pot, or should you fold and give up the pot to your opponent?
Remember your opponent has at least a pair of 10s. What cards could come up to make your hand better? What could come up to make your opponent’s hand better? Is there any way you could tie each other? Also look at the expected value of the game. If you fold, you are assured a loss of $24,000. Make sure this fact is used in your analysis.
Explain your strategy for solving the problem.
Explain why your strategy will work.
Execute your strategy showing your mathematical work.
Draw conclusions from your wor
To solve this problem, we need to analyze the situation and make a decision based on the probabilities and expected value.
1. Determine the possible hands:
- Your current hand: 6♦ 7♦
- Opponent's revealed hand: 10♣ 10♠
- Community cards: 3♦ 4♣ 9♠ 10♦
2. Assess your hand:
- With the current cards, you have an open-ended straight draw, meaning a 5 or 8 would complete a straight for you. There are 4 remaining 5s and 4 remaining 8s in the deck, so you have 8 outs to improve your hand.
3. Assess your opponent's hand:
- Your opponent has revealed a pair of 10s. The final card can't make them a better pair, but it could potentially give them three of a kind or a stronger hand.
4. Calculate the probabilities:
- The total number of unknown cards is 52 - 7 (2 in your hand, 4 on the board, and 1 revealed) = 45 cards.
- The probability of getting one of your outs on the river is 8/45 ≈ 0.178.
5. Evaluate the expected value:
- If you call, you can win the pot if you improve your hand or your opponent folds. The chance of winning the pot with your current hand is 0.178.
- If you fold, you lose the $24,000 you already bet.
Expected Value (EV) = (Probability of Winning) * (Pot size if you win) - (Probability of Losing) * (Amount you lose)
= (0.178) * ($48,000) - (1 - 0.178) * ($24,000)
= $8,544 - $19,392
= -$10,848
6. Analysis:
- The expected value of calling is negative (-$10,848). This means that on average, calling will result in a loss over time.
- Since you only have $10,000 left, you risk losing all of your remaining money if you call and lose the hand.
- Folding guarantees a loss of $24,000, less than the amount you would potentially lose by calling.
7. Conclusion:
- Based on the probabilities and expected value analysis, it is advisable to fold in this situation. Folding limits your loss to $24,000, while calling risks losing everything and has a lower expected value.