How do you calculate the probability of hitting a set with a pocket pair pre-flop?

How do you calculate the probability of hitting a set with a pocket pair pre-flop?

Correct Answer: B. Divide the number of desired outcomes by the number of possible outcomes (Correct)

Explanation:

The probability of hitting a set with a pocket pair pre-flop is calculated by dividing the number of desired outcomes (flopping a set) by the number of possible outcomes (the total number of flops). This probability is approximately 12%.

Example to Illustrate the Concept:

Scenario:

• Your Hand: Pocket pair, such as 8♠️ 8♣️.

Calculating the Probability:

1. Number of Desired Outcomes (Flopping a Set):
   • To hit a set on the flop, you need one of the remaining two 8s to appear on the flop.
   • The flop consists of three cards dealt from the remaining 50 cards in the deck.
   • The number of ways to choose 1 desired card (an 8) from the 2 remaining 8s and 2 non-desired cards from the remaining 48 cards:
     • The number of combinations to hit at least one 8 in the three-card flop can be calculated using combinatorial math.
2. Number of Possible Outcomes (Total Number of Flops):
   • The total number of ways to choose 3 cards from the remaining 50 cards is calculated using combinations.

Step-by-Step Calculation:

1. Calculate the Total Number of Flops:
   • The total number of ways to choose 3 cards from 50 cards: (503)=50!3!⋅(50−3)!=50×49×483×2×1=19,600\binom{50}{3} = \frac{50!}{3! \cdot (50-3)!} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 19,600(350​)=3!⋅(50−3)!50!​=3×2×150×49×48​=19,600
2. Calculate the Number of Desired Outcomes:
   • There are 2 remaining 8s and 48 other cards.
   • The number of ways to choose 1 of the 2 8s and 2 of the 48 other cards: Number of ways to choose 1 desired card from 2=(21)=2\text{Number of ways to choose 1 desired card from 2} = \binom{2}{1} = 2Number of ways to choose 1 desired card from 2=(12​)=2 Number of ways to choose 2 non-desired cards from 48=(482)=48×472×1=1,128\text{Number of ways to choose 2 non-desired cards from 48} = \binom{48}{2} = \frac{48 \times 47}{2 \times 1} = 1,128Number of ways to choose 2 non-desired cards from 48=(248​)=2×148×47​=1,128
   • Multiply these together to get the total number of desired outcomes: $$2\times 1,128=2,256$$
3. Calculate the Probability:
   • Divide the number of desired outcomes by the total number of possible outcomes: Probability=2,25619,600≈0.115=11.5%\text{Probability} = \frac{2,256}{19,600} \approx 0.115 = 11.5\%Probability=19,6002,256​≈0.115=11.5%

Practical Example:

Hand: 8♠️ 8♣️

• Action: You are dealt pocket 8s and want to know the probability of hitting a set on the flop.

Using the Calculation:

• Total Number of Flops: 19,600
• Number of Desired Outcomes: 2,256
• Probability: 11.5%

Conclusion:

• The probability of hitting a set with a pocket pair on the flop is approximately 11.5%, or roughly 1 in 9 times.

Benefits of Understanding the Probability:

1. Informed Decision-Making:
   • Example: Knowing the probability of hitting a set helps you make better decisions about whether to call a raise pre-flop with a pocket pair.
   • Outcome: Improved long-term profitability by making mathematically sound decisions.
2. Strategic Play:
   • Example: Understanding the low probability of hitting a set can help you manage expectations and avoid overcommitting chips with a pocket pair.
   • Outcome: Better bankroll management and reduced risk of significant losses.
3. Exploiting Opponents:
   • Example: Use your knowledge of probabilities to exploit opponents who overvalue their hands when you hit a set.
   • Outcome: Increased winnings when you do hit a set and can extract maximum value from your opponents.

Conclusion:

The probability of hitting a set with a pocket pair pre-flop is calculated by dividing the number of desired outcomes (flopping a set) by the number of possible outcomes (the total number of flops). This probability is approximately 12%. By understanding this probability, you can make more informed and strategic decisions in your poker play.