Question 1

Combinations

Accepted Answer

refer to the number of ways a subset of items can be selected from a larger set where the order of selection does not matter. Because a poker hand of Ace-King is identical to King-Ace, card probability relies heavily on combinations rather than Permutations (where order does matter). Outs is a practical term used primarily in poker, referring to the exact number of unseen cards remaining in the deck that will improve a player's hand to a winning hand. Pot Odds represents the ratio of the current size of the pot to the cost of a contemplated call. Finally, Expected Value (EV) is the mathematical average of all possible outcomes of a specific decision, weighted by the exact probability of each outcome occurring. A positive EV (+EV) indicates a mathematically profitable decision over time, while a negative EV (-EV) indicates a losing decision.

Question 2

What is the difference between probability and odds?

Accepted Answer

Probability is the mathematical ratio of favorable outcomes to the total number of possible outcomes, expressed as a fraction, decimal, or percentage (e.g., a 1 in 5 chance is 20%). Odds express the ratio of favorable outcomes to unfavorable outcomes (e.g., 1 win for every 4 losses is expressed as 1:4 or 4-to-1 odds). While they describe the same reality, odds are primarily used in gambling to quickly calculate payout ratios, whereas probability is used in pure statistics.

Question 3

Does shuffling the deck change the probabilities?

Accepted Answer

Assuming the deck is shuffled thoroughly and completely randomized, shuffling resets the sample space to its original state, but it does not change the fundamental probabilities of the deck's composition. The probability of drawing an Ace from the top of a shuffled deck is exactly the same as drawing an Ace from the bottom, or the exact middle. As long as the position of the cards is completely unknown to the observer, the mathematical probability of drawing any specific card remains constant at 1 in 52.

Question 4

What is a Monte Carlo simulation?

Accepted Answer

A Monte Carlo simulation is a computerized mathematical technique that relies on repeated random sampling to estimate complex probabilities. Instead of calculating the exact combinatorial math for a highly complex scenario (like a 5-way poker hand), the computer simply deals out the scenario randomly hundreds of thousands of times and tallies the results. Through the Law of Large Numbers, the average result of these massive simulations converges on the exact true mathematical probability.

Question 5

How do "blockers" affect card probability?

Accepted Answer

Blockers refer to known cards in your own hand or on the board that make it mathematically impossible (or less likely) for an opponent to hold a specific hand. For example, if you hold the Ace of Spades, you "block" your opponent from having the maximum possible hand of pocket Aces, because there are only 3 Aces left in the deck instead of 4. This reduces the combinations of pocket Aces your opponent can hold from 6 down to 3, cutting the probability in half based purely on the card you hold.

Question 6

Can knowing card probability guarantee a win?

Accepted Answer

No. Knowing card probability guarantees that you will make mathematically profitable decisions, but it cannot guarantee a win in any single, isolated event due to variance. If you have a 99% probability of winning a hand, mathematics dictates that you will still lose 1 out of every 100 times. Card probability is about maximizing Expected Value over an infinite timeline; it guarantees long-term profitability, not short-term immunity from bad luck.

Question 7

Why is the hypergeometric distribution used instead of the binomial distribution?

Accepted Answer

The binomial distribution is used for independent events where the probability of success remains constant on every trial (like flipping a coin). Card drawing without replacement is a dependent event; every time a card is drawn, the total population shrinks, and the ratio of remaining cards changes. The hypergeometric distribution is specifically designed for finite populations without replacement, making it the only accurate formula for modeling card games.

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