Problem 35 - Algebra and Trigonometry

Question

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, \( P(E \cup F) = \frac{11}{26} \). (b) Mutually exclusive, \( P(E \cup F) = \frac{1}{2} \).

Step 1: Check if the Events are Mutually Exclusive (a)

Events are mutually exclusive if they cannot occur at the same time. In case (a), event \(E\) is drawing a face card, and event \(F\) is drawing a spade. A face card can also be a spade, so they are not mutually exclusive.

Step 2: Calculate Probability of \(E \, \cap \, F\) (a)

The probability of \(E \, \cap \, F\) is the probability that a drawn card is both a face card and a spade. There are 3 face cards (Jack, Queen, King) in spades, so this probability is \( \frac{3}{52} \).

Step 3: Calculate Probability of \(E \, \cup \, F\) (a)

Use the formula for the probability of the union of two events: \(P(E \, \cup \, F) = P(E) + P(F) - P(E \, \cap \, F)\). There are 12 face cards, so \( P(E) = \frac{12}{52} \). There are 13 spades, so \( P(F) = \frac{13}{52} \). Thus, \( P(E \, \cup \, F) = \frac{12}{52} + \frac{13}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26} \).

Step 4: Check if the Events are Mutually Exclusive (b)

In case (b), event \(E\) is drawing a heart, while event \(F\) is drawing a spade. These two events cannot happen at the same time (a card cannot be both a heart and a spade), so they are mutually exclusive.

Step 5: Calculate Probability of \(E \, \cup \, F\) (b)

For mutually exclusive events, the probability of the union \(P(E \, \cup \, F)\) is simply \(P(E) + P(F)\). Since there are 13 hearts, \( P(E) = \frac{13}{52} \). Since there are 13 spades, \( P(F) = \frac{13}{52} \). Therefore, \( P(E \, \cup \, F) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} \).

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are two or more events that cannot happen at the same time. Imagine trying to draw one card that is both a heart and a spade from a deck; it's impossible, because no single card can be of two different suits.

In probability, identifying mutually exclusive events is crucial for simplifying calculations. When two events are mutually exclusive, the probability of their union is just the sum of their individual probabilities. In practical terms:

\[ P(E \cup F) = P(E) + P(F) \]

Understanding whether events are mutually exclusive helps in applying the right probability formula and ensures correct problem solving. For example, in our original exercise:

• (a) Face card and spade are not mutually exclusive. A face card can be a spade.
• (b) Heart and spade are mutually exclusive. A heart cannot be a spade.

Probability of Union

The probability of the union of two events, denoted as \(P(E \cup F)\), represents the likelihood of either event happening.

If events are not mutually exclusive, we must account for any overlap, where both events may happen simultaneously. The comprehensive formula for calculating the probability of union is:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

Here, \(P(E \cap F)\) accounts for the overlap, ensuring we don't double-count the probability where both events occur together.

In the context of a standard deck of cards, this understanding allows more accurate probability calculations. Consider event \(E\) as a face card and \(F\) as a spade; since face cards can be spades, \(P(E \cap F)\) is not zero and must be subtracted.

Face Card

Face cards in a deck are specific types of cards featuring pictures instead of numbers. These include the Jack, Queen, and King of each suit.

In a standard deck of 52 cards, there are 3 face cards in each of the four suits (hearts, diamonds, clubs, spades) making a total of 12 face cards.

• 3 face cards in hearts
• 3 face cards in diamonds
• 3 face cards in clubs
• 3 face cards in spades

The probability of drawing a face card from a standard deck is therefore: \[ P(E) = \frac{12}{52} \]
Knowing the specifics of face cards helps in determining probabilities, especially when combined with other events, like suits.

Standard Deck of Cards

A standard deck of cards is a commonly used set for many card games and probability exercises. It consists of 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, ranging from Ace through King.

The deck consists of

• 13 hearts,
• 13 diamonds,
• 13 clubs,
• 13 spades.

Understanding the structure of a standard deck is foundational for solving probability problems related to card games.

Each card has an equal chance of being drawn, and these equal probabilities make calculations straightforward when dealing with problems of randomness and unpredictability in card games. This basic knowledge lays the groundwork for recognizing probabilities, such as the chance of drawing any particular card or combination of cards.