Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

One card is selected at random from a standard deck of 52 playing cards. Use a formula to find the probability of the union of the two events. The card is a face card or a black card.

Step-by-Step Solution

Answer

The probability that the card drawn is either a face card or a black card is \( \frac{32}{52} \) or simplified, it's approximately 0.6154

Step 1: Understanding the Problem

A standard deck of cards has 52 cards. It includes 4 suits (hearts, diamonds, clubs, spades) with each suit having 13 cards (numbers 1-10 and the face cards king, queen, jack). Face cards are 12 in total (3 face cards * 4 suits). Black cards are either clubs or spades, making them 26 out of 52.

Step 2: Calculate Individual Probabilities

\[ P(Face\ Card) = \frac{Number\ of\ face\ cards}{Total\ cards} = \frac{12}{52} \]\n\[ P(Black\ Card) = \frac{Number\ of\ black\ cards}{Total\ cards} = \frac{26}{52} \]

Step 3: Calculate Overlapping Probability

The face cards that are also black are a King, Queen and, Jack of Spades and of Clubs, so 6 black face cards in total. Hence, \[ P(Face\ Card\ and\ Black\ Card) = \frac{Number\ of\ black\ face\ cards}{Total\ cards} = \frac{6}{52} \]

Step 4: Find Probability of Union of the Two Events

The union of two events A and B is given by \( P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B) \). Substituting our obtained probabilities: \[ P(Face\ Card\ or\ Black\ Card) = \frac{12}{52} + \frac{26}{52} - \frac{6}{52} = \frac{32}{52} \]

Key Concepts

Understanding Probability of Union

In probability theory, the union of two events, often denoted as \(P(A \cup B)\), represents the probability that either event \(A\) or event \(B\), or both occur. This is determined using a fundamental formula stemming from the principles of probability:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This formula considers the separate probabilities of each event and subtracts the probability of their overlap (\(P(A \cap B)\)) since it's counted twice when simply adding \(P(A)\) and \(P(B)\). This is crucial in problems where the events might not be mutually exclusive, meaning both events can occur simultaneously.
Think of it this way: if you're counting fruit, like apples and oranges, and you count all apples \(P(A)\) and add all oranges \(P(B)\), but you had some apple-oranges, you'd count those twice unless you subtract that overlap, \(P(A \cap B)\), to only count them once in the total of \(P(A \cup B)\).
For cards, this translates to us wanting to know how often we pull either a black card or a face card, accounting for cards that fit both categories.

Exploring a Standard Deck of Cards

A standard deck of playing cards is a common tool in both recreational games and in learning probability concepts. It consists of 52 cards divided among four suits: hearts, diamonds, clubs, and spades.
Each suit contains 13 cards: numbered cards ranging from 2 to 10, and three face cards known as the jack, queen, and king.

• Hearts and diamonds are red suits.
• Clubs and spades are black suits.

There are 26 red cards and 26 black cards in total. Recognizing the breakdown of these suits and their characteristics, such as the red and black and face cards, helps in calculating probabilities involving these cards.
It's a balanced, equal distribution among suits and card types making it a perfect classroom example. Understanding these elements is crucial for solving typical probability problems, such as the ones involving the union of different card types.

Identifying Overlapping Events in Probability

Overlapping events occur when two outcomes, or events, can happen simultaneously. In our card example, overlapping events are where cards are counted in both categories we're examining: face cards that are also black cards. These cards make up the intersection of the two events in our probability equation.
In the standard deck, both spades and clubs, being black suits, feature face cards (jack, queen, king). This makes them black face cards.

• There are three black face cards in spades.
• There are three black face cards in clubs.

Thus, there are six black face cards in total, meaning six cards are part of both the face card set and the black card set. When calculating the probability of drawing a face card or a black card, we must remember to account for these overlapping cards to avoid counting them twice which is why they are subtracted in the union probability formula. Understanding overlapping events helps in managing these overlaps and appropriately calculating true probabilities.