Problem 2 - Applied Discrete Structures

Question

Suppose that you are about to flip a coin and then roll a die. Let \(A=\) \(\\{H E A D S, T A I L S\\}\) and \(B=\\{1,2,3,4,5,6\\} .\) (a) What is \(|A \times B| ?\) (b) How could you interpret the set \(A \times B\) ?

Step-by-Step Solution

Answer

(a) The size of \(|A \times B|\) is 12. (b) \(A \times B\) represents all possible outcomes of flipping a coin and then rolling a die.

Step 1: Understand the Set

The set \(A\) represents the possible outcomes of a coin flip, which are \( \{HEADS, TAILS\} \). The set \(B\) represents the possible outcomes of rolling a die, which are \( \{1, 2, 3, 4, 5, 6\} \). Both sets describe independent events.

Step 2: Calculate the Size of the Cartesian Product

The Cartesian product \(A \times B\) is the set of all possible ordered pairs \((a, b)\) where \(a\) is in \(A\) and \(b\) is in \(B\). The size of the Cartesian product \(|A \times B|\) is equal to the product of the sizes of the individual sets \(|A|\) and \(|B|\). Since \(|A| = 2\) and \(|B| = 6\), the size of the Cartesian product is \(2 \times 6 = 12\).

Step 3: Interpret the Cartesian Product

The set \(A \times B\) can be interpreted as all possible outcomes of flipping a coin and then rolling a die. These outcomes are combinations of the results from each individual event. For example, some outcomes are \((HEADS, 1)\), \((HEADS, 2)\), \((TAILS, 1)\), etc., indicating the result of the coin and the result of the die.

Key Concepts

Probability Outcomes

When you flip a coin or roll a die, you are dealing with probability outcomes. Probability outcomes are all the possible results that can occur when you perform a random experiment.
In this exercise, the probability outcomes for the coin flip are simply "HEADS" and "TAILS."
Similarly, when rolling a six-sided die, the probability outcomes are "1," "2," "3," "4," "5," and "6."
These outcomes are each equally likely because the coin isn't biased, and the die is fair.
So, each outcome has the same chance of occurring.

• Coin flip outcomes: 2 options (HEADS or TAILS).
• Die roll outcomes: 6 options (1 through 6).

Putting them together, when you flip a coin and roll a die, you create a scenario of combined probability outcomes.
You will then think of all the possible combinations of these outcomes, which is where the Cartesian product comes into play.

Independent Events

Independent events are events where the occurrence of one does not affect the occurrence of another.
In this exercise, the outcome of flipping the coin has no impact on the outcome of rolling the die.
This means that the two events are independent of each other.

• The coin flip is independent of the die roll.
• The die roll is independent of how the coin lands.

This independence simplifies the way we can calculate the probability of composite outcomes.
This is because when events are independent, the probability of a combination of outcomes is just the product of the probabilities of each event occurring.
Independent events allow you to use the rule that combines probabilities simply by multiplying:
\[ P(A \text{ and } B) = P(A) \times P(B) \] Here, \(A\) and \(B\) could represent any outcomes of the connected events like the coin landing on HEADS and the die showing a 4.

Set Operations

Set operations are methods used to combine or relate different sets of data. One of these operations is the Cartesian product, which is denoted as \(A \times B\).
The Cartesian product is fundamental for creating connections between sets, especially in probability and statistics.

• In mathematical terms, the Cartesian product of two sets results in a set of all possible ordered pairs.
• The number of paired outcomes is the product of the number of members in each set.

This exercise involves computing the Cartesian product of \(A\) (the coin flip outcomes) and \(B\) (the die roll outcomes).
You find the ordered pairs by pairing each element of one set with every element of the other set.
By performing set operations like this, you can explore how different events can interact with one another in a comprehensive way.

Ordered Pairs

An ordered pair consists of two elements, often represented as \((a, b)\).
Here, the first component \(a\) comes from the first set and the second component \(b\) comes from the second set.
In this exercise, an ordered pair like \((HEADS, 3)\) indicates that the coin landed on "HEADS," and the die showed "3."

• First element: taken from the first set \(A\).
• Second element: taken from the second set \(B\).

It's important to note that the order matters in ordered pairs.
The first element is always from the first set, and the second element is always from the second set.
This concept helps ensure each outcome is distinct and meaningful.
Ordered pairs reflect real-world scenarios where sequences of events are important, like the result of flipping a coin followed by rolling a die.