Addition Rule: Videos & Practice Problems

Understanding the probability of multiple events is essential in probability theory, especially when distinguishing between events that can occur simultaneously and those that cannot. Events that cannot happen at the same time are termed mutually exclusive. For instance, if you consider the events of wearing a blue shirt (Event A) and wearing a green shirt (Event B), these events are represented by separate circles in a Venn diagram, indicating no overlap. This means you can either wear a blue shirt or a green shirt, but not both at the same time.

In contrast, events that can occur together are not mutually exclusive. For example, wearing a blue shirt (Event A) and wearing green pants (Event B) can happen simultaneously, as illustrated by overlapping circles in a Venn diagram. This overlap signifies that both events can occur at the same time.

To further clarify, consider flipping a coin. The outcomes of getting heads or tails are mutually exclusive since both cannot occur in a single flip. Conversely, when rolling a die, the event of rolling a six and the event of rolling a number higher than three are not mutually exclusive, as rolling a six satisfies both conditions.

When calculating the probability of mutually exclusive events, the approach is straightforward: you simply add the probabilities of each event. This is often referred to as or probability, denoted by the symbol ∪ in set notation. For example, if you want to find the probability of rolling a three or a five on a six-sided die, you would calculate it as follows:

Let \( P(A) \) be the probability of rolling a three, and \( P(B) \) be the probability of rolling a five. Since there is one way to roll a three and one way to roll a five, the probabilities are:

\[ P(A) = \frac{1}{6} \quad \text{and} \quad P(B) = \frac{1}{6} \]

Adding these probabilities gives:

\[ P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]

This means the probability of rolling a three or a five is \( \frac{1}{3} \), which is approximately 0.33. Understanding these concepts allows for a clearer grasp of how to approach problems involving multiple events in probability.

Understanding the calculation of the probability of two events occurring is essential in probability theory, especially when distinguishing between mutually exclusive and non-mutually exclusive events. For mutually exclusive events, the probability of either event A or event B occurring is straightforward: you simply add their individual probabilities together. However, when dealing with non-mutually exclusive events, an additional step is required due to the overlap where both events can occur simultaneously.

To calculate the probability of event A or event B for non-mutually exclusive events, you start by adding the probabilities of each event. However, since the overlap (where both events occur) has been counted twice, you must subtract the probability of that overlap. This can be expressed with the formula:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

In this formula, \( P(A \cup B) \) represents the probability of either event A or event B occurring, \( P(A) \) is the probability of event A, \( P(B) \) is the probability of event B, and \( P(A \cap B) \) is the probability of both events occurring simultaneously.

To illustrate this concept, consider the example of rolling a six-sided die. Let’s define two events: event A is rolling a number greater than three, and event B is rolling an even number. The possible outcomes when rolling the die are 1, 2, 3, 4, 5, and 6. Analyzing these outcomes:

• Event A (greater than 3): 4, 5, 6 (3 outcomes)
• Event B (even numbers): 2, 4, 6 (3 outcomes)
• Overlap (both greater than 3 and even): 4, 6 (2 outcomes)

Now, applying the formula:

1. Calculate \( P(A) \): There are 3 favorable outcomes (4, 5, 6) out of 6 total outcomes, so \( P(A) = \frac{3}{6} \).

2. Calculate \( P(B) \): There are also 3 favorable outcomes (2, 4, 6) out of 6 total outcomes, so \( P(B) = \frac{3}{6} \).

3. Calculate \( P(A \cap B) \): The overlap consists of 2 outcomes (4, 6), so \( P(A \cap B) = \frac{2}{6} \).

Substituting these values into the formula gives:

$$ P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{2}{6} = \frac{4}{6} = \frac{2}{3} $$

As a decimal, this probability is approximately 0.67. This result indicates that there are four favorable outcomes (2, 4, 5, 6) that satisfy either condition out of a total of six possible outcomes.

In summary, the general formula for calculating the probability of two events, whether they are mutually exclusive or not, remains consistent. For mutually exclusive events, the overlap term becomes zero, simplifying the calculation. This understanding allows for a comprehensive approach to probability, enabling the calculation of event probabilities in various scenarios.

In this scenario, we are tasked with determining the probability that a randomly selected individual from a group of 300 people is either wearing shorts or a green shirt. To approach this, we first identify the relevant data: 88 individuals are wearing shorts, and 106 are wearing a green shirt. Additionally, there are 89 individuals who are wearing both shorts and a green shirt, which illustrates the concept of non-mutually exclusive events.

To calculate the probability of either event occurring, we utilize the formula for the union of two events, which is expressed as:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Here, P(A) represents the probability of wearing shorts, P(B) represents the probability of wearing a green shirt, and P(A ∩ B) represents the probability of wearing both.

Calculating each component, we find:

• P(A) = Number of people wearing shorts / Total number of people = 88 / 300
• P(B) = Number of people wearing a green shirt / Total number of people = 106 / 300
• P(A ∩ B) = Number of people wearing both / Total number of people = 89 / 300

Substituting these values into the formula gives us:

P(A ∪ B) = (88/300) + (106/300) - (89/300)

Combining these fractions results in:

P(A ∪ B) = (88 + 106 - 89) / 300 = 105 / 300

This fraction simplifies to:

P(A ∪ B) = 41 / 60

Expressed as a decimal, this probability is approximately 0.68. Therefore, the likelihood that a randomly selected person from this group is wearing either shorts or a green shirt, or both, is 0.68, indicating a significant probability of this occurrence.