Question

Define the simple events for the experiments in Exercises \(16-20 .\) A single card is drawn from a deck of 52 playing cards.

Short answer

Answer: The simple events are the combination of all possible ranks and suits in the deck, consisting of 52 individual cards. These include the Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King of hearts, diamonds, clubs, and spades.

Step-by-step solution

Understand the composition of a deck of playing cards

A standard deck of playing cards has 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, which consist of the following ranks: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.

Define the simple events as individual cards

In this experiment, a simple event corresponds to drawing a specific card from the deck. Since there are 13 ranks and 4 suits, the total number of simple events is 13 x 4 = 52. Therefore, the simple events are the combination of all possible ranks and suits in the deck.

List all the simple events

List all 52 simple events by specifying the rank and suit of each card. The list of simple events will look like this: 1. Ace of hearts 2. 2 of hearts 3. 3 of hearts 4. 4 of hearts 5. 5 of hearts 6. 6 of hearts 7. 7 of hearts 8. 8 of hearts 9. 9 of hearts 10. 10 of hearts 11. Jack of hearts 12. Queen of hearts 13. King of hearts 14. Ace of diamonds 15. 2 of diamonds 16. 3 of diamonds 17. 4 of diamonds 18. 5 of diamonds 19. 6 of diamonds 20. 7 of diamonds 21. 8 of diamonds 22. 9 of diamonds 23. 10 of diamonds 24. Jack of diamonds 25. Queen of diamonds 26. King of diamonds 27. Ace of clubs 28. 2 of clubs 29. 3 of clubs 30. 4 of clubs 31. 5 of clubs 32. 6 of clubs 33. 7 of clubs 34. 8 of clubs 35. 9 of clubs 36. 10 of clubs 37. Jack of clubs 38. Queen of clubs 39. King of clubs 40. Ace of spades 41. 2 of spades 42. 3 of spades 43. 4 of spades 44. 5 of spades 45. 6 of spades 46. 7 of spades 47. 8 of spades 48. 9 of spades 49. 10 of spades 50. Jack of spades 51. Queen of spades 52. King of spades This list represents all the simple events for the experiment of drawing a single card from a deck of 52 playing cards.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes in experiments or processes that are random in nature. Think of it as the mathematical framework for quantifying uncertainty. It allows us to make predictions about the outcomes of various scenarios, even when they involve elements of chance.

For example, consider flipping a coin. There are two possible outcomes, heads or tails, and if the coin is fair, each outcome has an equal chance of occurring. Probability theory allows us to say that there's a 50% chance of getting heads and a 50% chance of getting tails. It extends these concepts to more complex situations, like the drawing of cards from a deck or rolling dice.

Deck of Playing Cards

A standard deck of playing cards is a common example used to understand probability theory. It typically consists of 52 cards, grouped into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, ranked from the Ace, which is often considered as either the highest or lowest card, to the King.

The individual ranks are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. By understanding the structure of a deck, you can calculate probabilities associated with various card games or exercises involving card drawing. A thorough comprehension of the makeup of a deck is crucial for accurate probability calculations in card-based statistical experiments.

Card Drawing Probability

Card drawing probability refers to the chances of drawing a specific card or combination of cards from a full or partial deck. When a single card is drawn at random from a standard 52-card deck, each card has an equal chance of being selected. This is an example of a uniform probability model, where the probability of choosing any card is \( \frac{1}{52} \) because there are 52 possible outcomes, each being a simple event.

These probabilities can become more complicated when considering conditional events, such as the likelihood of drawing a heart after already having drawn a few cards without replacing them. The act of card drawing in such a dynamic scenario demonstrates the idea of dependent events in probability.

Statistical Experiments

Statistical experiments are structured methods for collecting data, where the outcome is subject to randomness or uncertainty. The drawing of a card from a deck, as long as the card is drawn in a truly random fashion, is a classic example. Each draw of a card is an independent trial if the card is replaced back into the deck afterward.

Simple events in a statistical experiment are the most basic outcomes that cannot be further broken down. In the case of our card drawing, a simple event would be drawing the Queen of hearts, or the 2 of spades. The collection of all possible simple events forms what's known as the sample space in probability theory. By exploring these experiments, we can gain valuable insights into not only games of chance but also into more complex real-world processes where randomness plays a critical role.