Problem 44 - Algebra and Trigonometry

Question

\(43-44\) . 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: Define the Universal Set

The universal set for this problem is the set of all possible outcomes from rolling a die. This set is \( \{1, 2, 3, 4, 5, 6\} \) as these represent the possible faces of a standard six-sided die.

Step 2: Identify Elements of Event E in (a)

Event \( E \) for part (a) is that the number is greater than \(3\). Thus, the elements of \( E \) are \( \{4, 5, 6\} \).

Step 3: Identify Elements of Event F in (a)

Event \( F \) for part (a) is that the number is less than \(5\). Thus, the elements of \( F \) are \( \{1, 2, 3, 4\} \).

Step 4: Determine if E and F are Mutually Exclusive in (a)

Events are mutually exclusive if they have no elements in common. The intersection \( E \cap F \) is \( \{4\} \), indicating they are not mutually exclusive.

Step 5: Find Probability of E ∪ F in (a)

The union \( E \cup F \) includes outcomes \( \{1, 2, 3, 4, 5, 6\} \). The probability is calculated as \( \frac{6}{6} = 1 \).

Step 6: Identify Elements of Event E in (b)

Event \( E \) for part (b) is that the number is divisible by \(3\). Thus, the elements of \( E \) are \( \{3, 6\} \).

Step 7: Identify Elements of Event F in (b)

Event \( F \) for part (b) is that the number is less than \(3\). Thus, the elements of \( F \) are \( \{1, 2\} \).

Step 8: Determine if E and F are Mutually Exclusive in (b)

The intersection \( E \cap F \) is empty, \( \emptyset \), meaning they have no common elements and are mutually exclusive.

Step 9: Find Probability of E ∪ F in (b)

The union \( E \cup F \) includes outcomes \( \{1, 2, 3, 6\} \). The probability is calculated as \( \frac{4}{6} = \frac{2}{3} \).

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. When we think about this in terms of dice rolling, if one event happens, the other cannot. In other words, they have no common outcomes. For example:

• In exercise (a), we have two events:
  • Event \(E\): The number is greater than 3, which includes \( \{4, 5, 6\} \).
  • Event \(F\): The number is less than 5, which includes \( \{1, 2, 3, 4\} \).
    • Here, the intersection \(E \cap F\) is \( \{4\} \), so they are not mutually exclusive because they share the number 4.
  • In exercise (b), the events are:
    • Event \(E\): The number is divisible by 3, which includes \( \{3, 6\} \).
    • Event \(F\): The number is less than 3, which includes \( \{1, 2\} \).
      • Here, the intersection \(E \cap F\) is empty \( \emptyset \), so they are mutually exclusive because no number can fulfill both conditions.

Union of Events

The union of events in probability is an operation that combines two events, including any outcomes that are in either one of them or both. We denote union by \( E \cup F \). This is how we find all outcomes that satisfy at least one of the events.

For part (a) of our exercise:

• The union \(E \cup F\) of the events is \( \{1, 2, 3, 4, 5, 6\} \) because it covers all possible outcomes, both greater than 3 or less than 5, with probability \( \frac{6}{6} = 1 \).

For part (b) of our exercise:

• The union \(E \cup F\) is \( \{1, 2, 3, 6\} \), which includes numbers less than 3 or numbers that are divisible by 3, giving us a probability of \( \frac{4}{6} = \frac{2}{3} \).

Understanding the union helps in determining the probability of either or both events happening.

Die Rolling Outcomes

When rolling a standard six-sided die, you must consider all possible outcomes. This makes it easy to understand problems involving rolling dice.

• The universal set for a single roll of a six-sided die is \( \{1, 2, 3, 4, 5, 6\} \). Each number represents a possible face of the die. This set is the basis for determining probabilities involving die rolls.
• Each side has an equal chance of occurring, with a probability of \( \frac{1}{6} \) for each outcome.
• By understanding these outcomes, you can analyze events and their respective probabilities, such as finding mutually exclusive events and calculating the union of events, as seen in the exercises.

When considering events based on die rolls, breaking down the possible outcomes helps make sense of probability calculations.