Question

Draw three cards randomly from a standard deck of 52 cards and let \(x\) be the number of kings in the draw. Evaluate the probabilities and answer the questions in Exercises \(26-28\) Would the probability distribution in Exercise 27 change if \(x\) were defined to be the number of aces in the draw?

Short answer

In a standard 52-card deck, the probability distribution of drawing a certain number of kings (0, 1, 2, or 3) from a 3-card draw is the same as the probability distribution of drawing a certain number of aces. This is because there are the same number of kings and aces in a deck, and there is no change in the number of cards present in the deck that would affect the probabilities.

Step-by-step solution

Calculate the probability of drawing 0 kings

To calculate the probability of drawing 0 kings, we will choose 3 cards out of the 48 non-king cards and divide this by the total number of ways to choose 3 cards from a deck, which is \(_{52}C_{3}\): \(P(x=0) = \frac{_{48}C_{3}}{_{52}C_{3}}\)

Calculate the probability of drawing 1 king

To calculate the probability of drawing 1 king, we will choose 1 king from the 4 kings and 2 cards from the 48 non-king cards. Then we will divide this by the total number of ways to choose 3 cards from a deck, which is \(_{52}C_{3}\): \(P(x=1) = \frac{_{4}C_{1}\cdot_{48}C_{2}}{_{52}C_{3}}\)

Calculate the probability of drawing 2 kings

To calculate the probability of drawing 2 kings, we will choose 2 kings from the 4 kings and 1 card from the remaining 48 non-king cards. Then we will divide this by the total number of ways to choose 3 cards from a deck, which is \(_{52}C_{3}\): \(P(x=2) = \frac{_{4}C_{2}\cdot_{48}C_{1}}{_{52}C_{3}}\)

Calculate the probability of drawing all 3 kings

To calculate the probability of drawing 3 kings, we will choose all 3 kings from the 4 kings. Then we will divide this by the total number of ways to choose 3 cards from a deck, which is \(_{52}C_{3}\): \(P(x=3) = \frac{_{4}C_{3}}{_{52}C_{3}}\)

Compare the probability distribution for x as the number of aces

Since there are 4 kings and 4 aces in a standard 52-card deck, the probability distribution for \(x\) being the number of aces would be the same as the probability distribution for x as the number of kings. This is because the number of elements in each case is the same, and there is no change in the number of cards present in the deck that would affect the probabilities. Therefore, the probability distribution would not change if \(x\) were defined to be the number of aces in the draw.

Key concepts

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. It's essentially about counting, arranging, and grouping items in specific ways. For instance, if you want to know how many different ways you can draw three cards from a standard deck of 52, combinatorics provides the methods for calculating this.

One of these methods involves using combinations. A combination, denoted as \( _{n}C_{k} \) where \( n \) is the total number of items and \( k \) is the number selected, calculates the number of ways you can choose \( k \) items from a set of \( n \) without regard to the order. The formula for a combination is \( _{n}C_{k} = \frac{n!}{k!(n-k)!} \), where \( ! \) denotes a factorial, the product of all positive integers up to a given number.

In card games, counting combinations can help in figuring out the likelihood of drawing a specific hand. As an exercise improvement advice, it's essential to understand that the order of draw doesn't matter in such situations because a hand with the same cards is considered equivalent regardless of the order in which cards are drawn.

Probabilities in Card Games

In card games, probabilities are calculations used to determine the likelihood of drawing a certain hand or combination of cards. This is particularly important for strategizing in games like poker or to analyze the game's odds. In our exercise, we determine the probability of drawing 0, 1, 2, or 3 kings from a 52-card deck.

The probability is usually expressed as a ratio between the number of favorable outcomes over the total number of possible outcomes. For example, in step 1 of our exercise, the probability of drawing zero kings - denoted as \( P(x=0) \) - was calculated by dividing the number of ways to choose 3 cards from the 48 non-king cards by the total number of ways to choose 3 cards from the entire deck.

To visualize, imagine all possible three-card combinations laid out; those without kings are the ones favorable for \( P(x=0) \). Probabilities in card games often rely on understanding and computing such ratios, and being familiar with the deck's structure (4 suits, 2 colors, 13 ranks) can greatly assist in making these calculations.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent values under a given set of parameters. In the context of our card problem, these two values could be 'drawing a king' or 'not drawing a king'.

The distribution is defined by two parameters: \( n \) (the number of trials or draws) and \( p \) (the probability of success on an individual draw). The probability of getting exactly \( k \) successes (in our case, kings) in \( n \) independent trials is given by the formula: \( P(X = k) = _{n}C_{k} \times p^k \times (1-p)^{n-k} \).

In the exercise we reviewed, drawing cards would follow a binomial distribution if we repeated the draws multiple times. Each draw is independent (the outcome of one draw doesn't affect the others) and the probability of drawing a king remains constant if we replace the card each time. By understanding this concept, one can see that card draws can be viewed as a series of Bernoulli trials (yes/no experiments), which when summed up, create a binomial distribution.