Problem 7 - Algebra 2

Question

A card is drawn from a standard deck of cards. Determine whether the events are mutually exclusive or inclusive. Then find the probability. \(P(6 \text { or king })\)

Step-by-Step Solution

Answer

The events are mutually exclusive. The probability is \( \frac{2}{13} \).

Step 1: Understanding the Problem

We need to determine whether the events of drawing a '6' or a 'king' from a deck of cards are mutually exclusive or inclusive, and then find the probability of such an event occurring.

Step 2: Define Mutual Exclusivity

Two events are mutually exclusive if they cannot happen at the same time. In a deck of cards, drawing a '6' and drawing a 'king' are mutually exclusive because a card cannot be both a '6' and a 'king'.

Step 3: Calculate Total Possibilities

A standard deck of cards has 52 cards. Each card is a unique combination of rank and suit.

Step 4: Determine Favorable Outcomes for '6'

There are four '6s' in each of the suits: hearts, diamonds, clubs, and spades. So, there are 4 possible outcomes for drawing a '6'.

Step 5: Determine Favorable Outcomes for 'King'

Similarly, there are four 'kings' in each of the suits: hearts, diamonds, clubs, and spades. So, there are 4 possible outcomes for drawing a 'king'.

Step 6: Count Combined Favorable Outcomes

Since '6' and 'king' are mutually exclusive, their outcomes do not overlap. Combine the outcomes for '6' (4) and 'king' (4) to get a total of 8.

Step 7: Calculate Probability

To find the probability of drawing either a '6' or a 'king', divide the total number of favorable outcomes by the total number of cards. So, the probability is \( \frac{8}{52} \) which simplifies to \( \frac{2}{13} \).

Key Concepts

Mutually Exclusive Events

In probability theory, two events are said to be mutually exclusive if they cannot both occur at the same time. For instance, in a single draw from a standard deck of cards, you cannot draw a card that is simultaneously a '6' and a 'king'. This characteristic is what makes the events mutually exclusive.
When dealing with mutually exclusive events, the probability of either event happening is simply the sum of their individual probabilities. Therefore, for mutually exclusive events, the intersection of the two events is always zero, as they cannot happen together. This simplifies calculations, as seen in our exercise, where drawing a '6' and drawing a 'king' are mutually exclusive.

Standard Deck of Cards

A standard deck of cards is an essential tool in probability exercises, comprising 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 ranks: numbers 2 through 10, plus the jack, queen, king, and ace.
Understanding the composition of a deck is crucial because it allows you to quickly determine the total number of possible outcomes in any card-drawing scenario. This consistency in structure is why the standard deck is ubiquitous in discussing probability concepts, as it provides a straightforward example of how to calculate chances and outcomes.

Probability Calculation

Probability is the measure of how likely an event is to occur, expressed as a fraction or a decimal between 0 and 1. To calculate probability, use the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
For our exercise, we wanted to find the probability of drawing either a '6' or a 'king'. By recognizing these as mutually exclusive events, we added the separate probabilities of drawing each:

• Favorable outcomes for drawing a '6': 4
• Favorable outcomes for drawing a 'king': 4
• Total number of possible outcomes (cards in the deck): 52

Therefore, the probability is:
\( P(6 \text{ or king}) = \frac{4 + 4}{52} = \frac{8}{52} = \frac{2}{13} \)

Favorable Outcomes

Favorable outcomes are the specific results from a probability event that we are interested in occurring. These are defined based on the criteria of the problem. In the card-drawing exercise, the favorable outcomes are the situations where a '6' or a 'king' is drawn.
Identifying favorable outcomes requires knowing both the structure of the set you are working with (like a deck of cards) and the specific event you are measuring. Here, each suit contains one '6' and one 'king', yielding four favorable outcomes for '6' and four for 'king'. Since these outcomes do not overlap under the category of mutually exclusive events, you simply add them to find a total of eight favorable outcomes in drawing either a '6' or a 'king'.