Problem 45 - Algebra and Trigonometry

Question

\(45-46\) A card is drawn at random from a standard 52 -card deck. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The card is a face card. F: The card is a spade. (b) \(E :\) The card is a heart. F: The card is a spade.

Step-by-Step Solution

Answer

(a) Not mutually exclusive; P(E∪F)=11/26. (b) Mutually exclusive; P(E∪F)=1/2.

Step 1: Understanding Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time. That is, if one event occurs, the other cannot occur.

Step 2: Defining Event E and Event F for Part (a)

For part (a), event E is that the card is a face card. There are 12 face cards in a deck: 4 Jacks, 4 Queens, and 4 Kings. Event F is that the card is a spade. There are 13 spades in total.

Step 3: Checking for Mutually Exclusiveness in Part (a)

Check if there is any card that can be both a face card and a spade. There are three spade face cards: Jack, Queen, and King of Spades. Since both events can occur simultaneously, E and F are not mutually exclusive.

Step 4: Calculating P(E ∪ F) for Part (a)

To find the probability of either E or F occurring, use the formula: \( P(E \cup F) = P(E) + P(F) - P(E \cap F) \). Here, \( P(E) = \frac{12}{52}, P(F) = \frac{13}{52}, \) and \( P(E \cap F) = \frac{3}{52} \). Thus, \( P(E \cup F) = \frac{12}{52} + \frac{13}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26} \).

Step 5: Defining Event E and Event F for Part (b)

For part (b), event E is that the card is a heart (13 hearts total in a deck), and event F is that the card is a spade.

Step 6: Checking for Mutually Exclusiveness in Part (b)

Check if there is any card that can be both a heart and a spade. There is no such card, meaning events E and F are mutually exclusive.

Step 7: Calculating P(E ∪ F) for Part (b)

Since E and F are mutually exclusive, use the formula \( P(E \cup F) = P(E) + P(F) \). Thus, \( P(E \cup F) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} \).

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are essential in probability theory. They describe situations where two events cannot occur simultaneously. If we take a closer look, this means that the occurrence of one event excludes the possibility of the other happening at the same time.
For example, consider drawing a card from a deck. If you draw a heart, you cannot simultaneously draw a spade. These two outcomes cannot both happen in a single draw and thus, they are mutually exclusive events.
To determine if events are mutually exclusive, identify any outcomes that could satisfy both events. If there aren't any, the events do not overlap, meaning they are mutually exclusive. When calculating probabilities with mutually exclusive events, the formula simplifies since no overlap exists. It's important to remember that mutually exclusive events dramatically affect how probabilities are managed in calculations.

Card Deck Probability

A standard deck of cards contains a fixed set of 52 cards, presenting a straightforward model for studying probability. Each card in the deck is unique in terms of suit and rank, allowing for clear calculations.
Key characteristics of a standard deck include:

• 4 suits: hearts, diamonds, clubs, and spades
• 13 ranks in each suit: Ace through 10, plus Jack, Queen, and King
• 12 face cards total: 4 Jacks, 4 Queens, 4 Kings

Understanding the structure of the deck is crucial to calculating probabilities accurately. For instance, to find the probability of drawing a face card, you'd consider there are 12 face cards out of 52, thus the probability is \( \frac{12}{52} \).
When dealing with problem situations, these basic probabilities can be expanded. By understanding the total number of cards and specific characteristics, one can easily build and solve complex probability scenarios within a deck.

Union of Events

The union of events refers to a combined scenario where any one of several events could occur. In probability, finding the union often involves calculating the likelihood of at least one event from a set occurring.
A key formula used is: \[P(E \cup F) = P(E) + P(F) - P(E \cap F)\]This formula accounts for overlapping probabilities, ensuring that probabilities aren't simply added up and inflated. When there is no overlap—as in the case of mutually exclusive events—this simplifies to just adding probabilities directly: \( P(E \cup F) = P(E) + P(F) \).
Understanding the union of events is crucial, especially in scenarios where multiple outcomes are possible, and some outcomes can satisfy more than one event. This ensures accurate probability calculations, avoiding overestimation or underestimation.