Problem 21 - Algebra 2

Question

For Exercises \(20-23,\) determine whether the events are mutually exclusive or inclusive. Then find the probability. A die is rolled. What is the probability of rolling a 5 or a number greater than 3\(?\)

Step-by-Step Solution

Answer

Events are inclusive; probability is \( \frac{2}{3} \).

Step 1: Understand the Events

We have two events here. Let's define them: Event A is rolling a 5. Event B is rolling a number greater than 3. The outcomes on a single roll can be 1, 2, 3, 4, 5, or 6.

Step 2: Determine If Events Are Mutually Exclusive or Inclusive

Events are mutually exclusive if they cannot happen at the same time. Event A (rolling a 5) and Event B (rolling a number greater than 3) are inclusive since rolling a 5 is also part of rolling a number greater than 3 (Event B includes 4, 5, and 6).

Step 3: Calculate Probability for Event A

The probability of rolling a 5 (Event A) is calculated using the formula \( P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total possible outcomes}} = \frac{1}{6} \).

Step 4: Calculate Probability for Event B

The probability of rolling a number greater than 3 (Event B includes outcomes 4, 5, and 6) is calculated as \( P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total possible outcomes}} = \frac{3}{6} = \frac{1}{2} \).

Step 5: Calculate Probability of A and B

The probability of both events happening (rolling a 5, which is common in both A and B) is \( P(A \cap B) = \frac{1}{6} \).

Step 6: Apply the Addition Rule of Probability

Since the events are inclusive, we apply the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Substitute the known values: \( \frac{1}{6} + \frac{1}{2} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \).

Step 7: Solution Conclusion

The probability of rolling a 5 or a number greater than 3 is \( \frac{2}{3} \).

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. In other words, if one event occurs, the other cannot. Think of it like choosing an ice cream flavor; you can either have chocolate or vanilla, but not both at the same time.
For example:

• If you flip a coin, landing on heads means you cannot also land on tails. These outcomes are mutually exclusive.
• Drawing a single card from a deck, drawing an Ace of Spades, and an Ace of Hearts are mutually exclusive since one card draw cannot yield both.

In probability, understanding whether events are mutually exclusive helps us decide how to calculate the combined probability of these events.

Inclusive Events

Inclusive events, unlike mutually exclusive events, can occur at the same time. This overlap means that one event does not prevent the occurrence of another.
In the context of rolling a die:

• Rolling a 5 and rolling a number greater than 3 are inclusive events. This is because rolling a 5 is also part of the outcomes when rolling a number greater than 3 (which includes 4, 5, and 6).
• Another example is the probability of drawing a red card or a heart from a deck of cards. These events overlap because all hearts are red.

Understanding inclusive events is crucial, as it influences how we apply the rules of probability to calculate the chance of one or more events occurring.

Addition Rule of Probability

The Addition Rule of Probability is a formula used to find the probability of either one event or another occurring. When events are inclusive, meaning they can happen at the same time, we need to adjust how we calculate probabilities to avoid overcounting.
The formula is:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

• \(P(A \cup B)\) denotes the probability of either event A or event B occurring.
• \(P(A)\) is the probability of event A occurring.
• \(P(B)\) is the probability of event B occurring.
• \(P(A \cap B)\) is the probability of both events A and B occurring, correcting for any overlap.

For non-mutually exclusive events like rolling a 5 or a number greater than 3, we use the inclusive version of the rule to accurately calculate the combined probability of the events.

Outcome Space

The outcome space, also known as the sample space, is a fundamental concept in probability. It refers to the set of all possible outcomes of an experiment.

Consider when rolling a standard six-sided die:

• The outcome space is {1, 2, 3, 4, 5, 6} because these are all the numbers the die can land on.
• In card games, the outcome space of drawing one card from a standard deck is all 52 cards.

Understanding the outcome space is critical when determining probabilities, as it gives us the total number of possible results against which favorable events are compared. The probability of any event can be calculated by dividing the number of favorable outcomes by the total number of outcomes in the outcome space.