Problem 34 - Algebra and Trigonometry

Question

A die is rolled and the number showing is observed. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The number is greater than 3 \(F :\) The number is less than 5 (b) \(E :\) The number is divisible by 3 \(F :\) The number is less than 3

Step-by-Step Solution

Answer

(a) Not mutually exclusive, \( P(E \cup F) = 1 \). (b) Mutually exclusive, \( P(E \cup F) = \frac{2}{3} \).

Step 1: List Possible Outcomes for Event E and F - Part (a)

For event E, which is the number greater than 3, the possible outcomes when a die is rolled are {4, 5, 6}. For event F, which is the number less than 5, the possible outcomes are {1, 2, 3, 4}. These are on a standard 6-sided die with faces numbered from 1 to 6.

Step 2: Determine the Intersection of E and F - Part (a)

To find if events E and F are mutually exclusive, check for common outcomes. Intersect the sets of E and F: {4, 5, 6} ∩ {1, 2, 3, 4} = {4}. Since they have a common element, they are not mutually exclusive.

Step 3: Calculate Probability of E ∪ F - Part (a)

The union E ∪ F represents outcomes either greater than 3 or less than 5, which translates to the set {1, 2, 3, 4, 5, 6}. Since every possible outcome of rolling a die is covered, the probability of E ∪ F is 1.

Step 4: List Possible Outcomes for Event E and F - Part (b)

Event E is numbers divisible by 3, leading to outcomes {3, 6}. Event F is numbers less than 3, leading to outcomes {1, 2}. These represent the outcomes on a 6-sided die.

Step 5: Determine the Intersection of E and F - Part (b)

Intersect the possible outcomes of E and F for mutual exclusivity. {3, 6} ∩ {1, 2} = {} (empty set). Since no outcomes overlap, E and F are mutually exclusive.

Step 6: Calculate Probability of E ∪ F - Part (b)

The union E ∪ F is outcomes either divisible by 3 or less than 3, which gives {1, 2, 3, 6}. The probability comes from counting successful outcomes which total 4, from a possible 6 (the sides of a die), leading to \( P(E \cup F) = \frac{4}{6} = \frac{2}{3} \).

Key Concepts

Mutually Exclusive Events

In probability, events are called mutually exclusive if they cannot occur at the same time. This means if one event happens, the other cannot. For example, when rolling a single six-sided die, the event of rolling an even number (2, 4, 6) is mutually exclusive with rolling an odd number (1, 3, 5), as these outcomes cannot both happen at once.

In the provided exercise:

• For part (a), events E (number greater than 3) and F (number less than 5) are not mutually exclusive because they share a common outcome, which is 4.
• For part (b), events E (number divisible by 3) and F (number less than 3) are mutually exclusive because no number is both in E and F.

Understanding if events are mutually exclusive is crucial for calculating probabilities accurately.

Union of Events

The union of two events in probability, denoted as \(E \cup F\), includes all outcomes that are in either event or both. Essentially, it's about finding the probability that at least one of the events occurs when the experiment is performed.

In the exercise, we determine the union for two sets of events:

• In part (a), the union includes numbers either greater than 3 or less than 5, which are {1, 2, 3, 4, 5, 6}. Hence, every outcome of rolling a die is included, making the probability of \(E \cup F = 1\).
• In part (b), the union consists of numbers divisible by 3 or less than 3, resulting in outcomes {1, 2, 3, 6}. Thus, the probability is \( \frac{4}{6} = \frac{2}{3} \).

This concept is widely used in probability, aiding in the calculation of combined outcomes from separate events.

Rolling a Die

Rolling a die is a classic example in probability studies. A standard die has 6 faces, numbered from 1 to 6, providing a simple way to understand basic concepts of probability.

Each roll represents a random experiment where each number has an equal chance of appearing. The outcomes are:

• Number 1
• Number 2
• Number 3
• Number 4
• Number 5
• Number 6

When exploring different events with a die, you can easily calculate probabilities as each side represents an outcome with a probability of \( \frac{1}{6} \). This straightforward scenario helps reinforce key ideas like sample space and event calculations.

Divisibility

Divisibility refers to determining whether one number can be divided by another without leaving a remainder. In probability, this concept helps define specific events.

• A number is divisible by 3 if, when divided by 3, the result is a whole number. On a die:
  • 3 and 6 are divisible by 3.
• In part (b) of the exercise, event E refers to numbers divisible by 3, with outcomes {3, 6}.

Understanding divisibility aids in grouping specific outcomes together when calculating probabilities and creating clear event conditions.

Outcome Space

The outcome space, or sample space, is the set of all possible outcomes in a probabilistic scenario. For a six-sided die, this is typically: {1, 2, 3, 4, 5, 6}. This space represents every possible result an event can produce in an experiment.

In probability:

• Understanding the complete outcome space is essential as it provides the basis for calculating probabilities for different events.
• For any event, probabilities are calculated by considering the number of favorable outcomes over the total possible outcomes.

Always start by defining the outcome space clearly. It helps you determine the likelihood of specific events accurately and ensures you don't overlook any potential results. Understanding the outcome space is foundational for all probability computations.