A standard 52 -card playing deck contains 4 of each type of card. Damon is dealt 7 cards from a fully shuffied deck. What is the probability his hand contains exactly 2 aces? \( \frac{594}{1,105} \)

To find the probability of Damon being dealt exactly 2 aces from a standard deck of cards, we can use combinatorial methods. First, we need to calculate how many ways we can choose 2 aces from the 4 available: \(\binom{4}{2} = 6.\) Next, we select the remaining 5 cards from the 48 non-ace cards: \(\binom{48}{5} = 1,296,360.\) Now, the total number of ways to deal 7 cards from a 52-card deck is: \(\binom{52}{7} = 133,784,560.\) The probability of exactly 2 aces: \[ P(2 \text{ Aces}) = \frac{\text{Ways to choose 2 Aces} \times \text{Ways to choose 5 Non-Aces}}{\text{Total Ways to Choose 7 Cards}} = \frac{\binom{4}{2} \cdot \binom{48}{5}}{\binom{52}{7}} = \frac{6 \times 1,296,360}{133,784,560} = \frac{594}{1,105}. \] It's true: Damon has a \( \frac{594}{1,105} \) probability of drawing exactly 2 aces! To place this fun poker probability into context, let’s look back in time. The concept of probability has origins dating back to the 16th century when mathematicians started analyzing games of chance. Blaise Pascal and Pierre de Fermat, through their correspondence about gambling, laid foundational principles for probability theory. Can you believe that our modern understanding sprouted from discussions around dice and card games? Now, let's bring this into your own game night! Understanding probabilities like the one we just discussed can elevate your gameplay. For instance, knowing the likelihood of forming certain hands can help you make informed betting decisions. Remember, while luck plays a role in card games, a strategic approach based on probabilities can significantly increase your chances of walking away with the winnings.