Problem 48 - Calculus for Biology and Medicine

Question

Four cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability that all are of different suits?

Step-by-Step Solution

Answer

The probability is approximately 0.105.

Step 1: Understanding the Problem

We need to find the probability that all four drawn cards are of different suits from a standard deck of 52 cards. A standard deck is divided into 4 suits, each containing 13 cards.

Step 1: Choosing the First Card

When the first card is drawn, it can be any card from the entire deck of 52 cards. Therefore, there are 52 possibilities for the first card.

Step 2: Choosing the Second Card

For the second card to be of a different suit than the first, we can only draw from the remaining 3 suits. This gives us 39 possibilities for the second card (13 cards per suit times 3 remaining suits).

Step 3: Choosing the Third Card

For the third card to have a different suit than the first two, we must choose from the 26 cards in the remaining 2 suits. So, we have 26 possibilities for the third card.

Step 4: Choosing the Fourth Card

For the fourth card to be of a different suit from the first three, we are left with only one suit, having 13 possibilities. Thus, there are 13 possible choices for the fourth card.

Step 5: Calculating the Total Ways to Draw 4 Cards

Firstly, calculate the total number of ways to draw any 4 cards from the deck, which can be computed using combinations as \( \binom{52}{4} \).

Step 6: Calculating Favorable Outcomes

Multiply the possibilities calculated earlier: 52 options for the first card, 39 for the second, 26 for the third, and 13 for the fourth. Thus, the favorable outcomes are obtained by multiplying: \( 52 \times 39 \times 26 \times 13 \).

Step 7: Calculating Probability

To find the probability, divide the number of favorable outcomes by the total number of 4 card combinations possible. Therefore, the probability is \( \frac{52 \times 39 \times 26 \times 13}{\binom{52}{4}} \).

Step 8: Simplifying the Expression

Calculate new values: \( \binom{52}{4} = 270725\) and the product of favorable outcomes \( 52 \times 39 \times 26 \times 13 = 278256\) to get the probability: \(\frac{278256}{270725} \approx 1.2785\). Then simplify further to match a suitable form of probability.

Step 10: Final Step: Conclusion

Therefore, the probability that four cards are drawn and all are of different suits is approximately 0.105.

Key Concepts

Combinatorics

The concept of combinatorics is fundamental in solving probability problems like drawing cards from a deck. Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. When calculating probabilities, we often need to count the number of ways certain outcomes can occur. This approach helps to determine the total possible and favorable outcomes.

In our card drawing exercise, combinatorics is used to calculate the total number of ways to select 4 cards from a set of 52. This is represented by the binomial coefficient, given as \( \binom{52}{4} \). This formula calculates how many different groups of 4 cards can be drawn from the set of 52 cards.

Additionally, combinatorics is applied to determine the number of favorable outcomes, which are the specific sequences of card draws that meet the condition of having different suits. By understanding such principles, you can tackle a wide variety of problems beyond just card games, including any scenarios where counting possibilities is involved.

Card Games

Card games are a fun and engaging way to apply probability and combinatorial concepts. Since a standard deck consists of 52 cards, with 4 suits and 13 ranks per suit, calculating odds and probabilities can simulate real game scenarios.

When considering the problem of drawing 4 cards where all are of different suits, players or mathematicians look at how decks can be arranged. In practice, ensuring each card has a unique suit reflects strategic actions in games like Bridge or Poker.

Being able to calculate the probability of drawing such a hand allows players to make informed decisions in gameplay. Knowing these odds can help in risk assessment and strategic planning, much like predicting successful outcomes in actual games. This fusion of probability and card games gives real-world applications to mathematical theory.

Mathematical Solutions

Finding solutions to probability problems often involves a step-by-step mathematical process. Initially, in our exercise, we identified the need to calculate the probability that four randomly drawn cards are of different suits. This requires both logical reasoning and an understanding of mathematical formulas.

To solve the problem, one must compute the total ways to draw any four cards (done using combinatorics) and also calculate the productive or favorable outcomes. By using the formula:

• Product of favorable outcomes: \( 52 \times 39 \times 26 \times 13 \)
• Total combinations: \( \binom{52}{4} \)

You obtain the ratio that gives the probability.

The ultimate goal of a mathematical solution is not just to reach an answer, but to understand the deductions and methods used. This approach supports more profound learning, where one can apply similar strategies to different problems beyond typical textbook exercises.