Question

Draw five cards randomly from a standard deck of 52 cards, and let \(x\) be the number of red cards in the draw. Evaluate the probabilities in Exercises \(22-25\). \(P(x=5)\)

Short answer

Answer: The probability of drawing five red cards from a standard deck of 52 cards is approximately 0.0253 or 2.53%.

Step-by-step solution

Identify the relevant parameters

In our case, we have: - \(N = 52\) (total number of cards in the deck) - \(n = 26\) (total number of red cards in the deck) - \(r = 5\) (number of cards drawn) - \(k = 5\) (number of red cards drawn)

Calculate the combinations

Next, we need to calculate the combination values using the binomial coefficient formula: 1. \(\binom{n}{k} = \binom{26}{5}\) - the number of ways to choose 5 red cards from the 26 available. 2. \(\binom{N-n}{r-k} = \binom{52-26}{5-5} =\binom{26}{0}\) - the number of ways to choose 0 black cards from the 26 available. 3. \(\binom{N}{r} = \binom{52}{5}\) - the number of ways to choose 5 cards from the total deck of 52 cards.

Apply the binomial probability formula

Now we can insert these values into the binomial probability formula to calculate \(P(x=5)\): $$P(x=5) = \frac{\binom{26}{5}\binom{26}{0}}{\binom{52}{5}}$$

Evaluate the combinations

Using the combination formula, we can evaluate each combination value as follows: 1. \(\binom{26}{5} = \frac{26!}{5!(26-5)!} = \frac{26!}{5!21!} = 65,780\) 2. \(\binom{26}{0} = \frac{26!}{0!(26-0)!} = 1\) (since choosing 0 elements from a set results in just one choice: the empty set) 3. \(\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = 2,598,960\)

Compute the final probability

Substitute the combination values back into the probability formula: $$P(x=5) = \frac{65,780\times1}{2,598,960} = \frac{65,780}{2,598,960}$$ Now, simplify the fraction to obtain the final probability: $$P(x=5) = \frac{65,780}{2,598,960} \approx 0.0253$$ So, the probability of drawing five red cards from a standard deck of 52 cards is approximately \(0.0253\) or \(2.53\%\).

Key concepts

Binomial Coefficient

Understanding the binomial coefficient is crucial when dealing with probabilities in card games. In a deck of cards, you may be asked about the likelihood of drawing a particular combination of cards. The binomial coefficient, symbolized as \( \binom{n}{k} \), tells us the number of ways to choose \( k \) items from a larger set of \( n \) items without considering the order of selection.

For example, if you have 26 red cards in a deck and you want to find out in how many ways you can draw 5 red cards, you would use the binomial coefficient formula: \( \binom{26}{5} \). This calculation would tell you the total number of possible 5-card combinations that can be drawn from the 26 red cards. With factorials involved in the formula, where \( n! \) denotes the factorial of \( n \) which is the product of all positive integers up to \( n \) (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)), we get a clear picture of the total combinations possible. Understanding this concept is essential for calculating probabilities in card games and beyond.

Combination Formula

The combination formula expands on the concept of the binomial coefficient by providing the mathematical means to determine the number of possible combinations in a given situation. It is represented as \( C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This formula assumes that the order of selection does not matter, which is often the case in card games like poker or when drawing cards from a deck.

Let's take the example of drawing cards from a deck. If you wish to draw 5 cards from the 52 available in the deck, the combination formula helps you calculate the total number of possible 5-card hand combinations regardless of their order. This is done by applying the formula: \( \binom{52}{5} \. Using the solution steps, we can see that \( \binom{52}{5} = 2,598,960 \), indicating there are 2,598,960 unique ways to draw a 5-card hand from a 52-card deck.

Probability Formula

The probability formula is what actually allows us to calculate the likelihood of a particular event occurring. For drawing cards, the basic probability formula is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). When calculating the probability of drawing five red cards from a deck, we consider the number of ways to achieve this outcome (favorable outcomes) against the total number of 5-card combinations possible (total outcomes)

. Using the combination values obtained through binomial coefficients, we can plug these into the probability formula to determine the exact likelihood. For drawing five red cards from a deck, we have \( P(x=5) = \frac{65,780}{2,598,960} \) after evaluating the combination numbers. Simplifying this fraction gives us the probability, which in the example provided is approximately 2.53%. By mastering the probability formula, students can tackle a variety of probability problems beyond card games, including those in different aspects of mathematics and real-world scenarios.