Problem 41 - 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 club or a king.

Step-by-Step Solution

Answer

The probability of drawing a card that is either a club or a king is \(\frac{6}{13}\).

Step 1: Calculate the Probability of Drawing a Club

Firstly, calculate the probability of drawing a club. Clubs constitute 13 out of 52 cards in a deck, equivalently \(\frac{1}{4}\). So, the probability \(P(C)\) of drawing a club will be \(\frac{13}{52}=\frac{1}{4}\).

Step 2: Calculate the Probability of Drawing a King

Next, calculate the probability of drawing a king. There are 4 kings in a standard deck of 52 cards, equivalently \(\frac{1}{13}\). So, the probability \(P(K)\) of drawing a king will be \(\frac{4}{52}=\frac{1}{13}\).

Step 3: Account for Overlap Between Events

One king is also a club, which represents overlap between the two events. Because this card has been counted twice, once in \(P(C)\) and once in \(P(K)\), it is necessary to subtract this overlapping card to ensure it is not counted twice. The probability \(P(C \cap K)\) that this card is drawn is \(\frac{1}{52}\).

Step 4: Calculate the Probability of the Union of the Two Events

Finally, use the formula \(P(C \cup K) = P(C) + P(K) - P(C \cap K)\) to calculate the probability of the union of the two events, which is \(\frac{1}{4} + \frac{1}{13} - \frac{1}{52} = \frac{12}{52} = \frac{6}{13}\).

Key Concepts

Union of Events

Probability theory often deals with multiple events occurring at once. When analyzing these, the **union of events** concept becomes important. The union represents the event where at least one of the events occurs. For example, if we have two events, A and B, their union is denoted by \(A \cup B\), meaning "A or B or both."

To compute the probability of the union of two events, we use the formula:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

This formula ensures that if both events can happen simultaneously, we do not double-count their intersection, \(P(A \cap B)\), which represents the probability of both events A and B occurring.

Understanding the calculation of the union helps avoid errors in probability problems, especially when events overlap.

Standard Deck of Cards

A standard deck of playing cards consists of 52 cards. This deck is divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, and these include number cards (2 through 10) and face cards (jack, queen, king, and ace).

For anyone working on probability problems involving cards, it's crucial to have a clear understanding of this structure. Here are key characteristics to remember:

• There are exactly 4 of each card rank (e.g., four 7s, four kings) in different suits.
• Hearts and diamonds are red-colored, while clubs and spades are black-colored.
• In total, there are 12 face cards, composed of 4 jacks, 4 queens, and 4 kings.

This simple structure allows for a range of probability scenarios and calculations, such as finding the probability of drawing a specific suit or rank, or more complex events like those with overlapping characteristics.

Overlapping Events

Overlapping events in probability occur when events can happen simultaneously. This means that there is a common outcome in the two events. This is important in probability problems because it affects how we compute the overall probability.

Using the deck of cards example, the overlap between drawing a club and drawing a king is the "King of Clubs." Thus:

• Probability of drawing a club, \(P(C) = \frac{13}{52}\).
• Probability of drawing a king, \(P(K) = \frac{4}{52}\).
• Probability of the overlap, drawing the "King of Clubs," \(P(C \cap K) = \frac{1}{52}\).

In problems with overlapping events, deducting the overlap from the total probabilities ensures accurate results. By adjusting for overlaps in calculations, we take into account the shared outcomes that have been double-counted when adding the individual probabilities. This adjustment is fundamental to accurately determining the probabilities of events that can occur at the same time.