Problem 18 - Basic Statistics for Business and Economics with Student

Question

Let \(P(X)=.55\) and \(P(Y)=.35 .\) Assume the probability that they both occur is .20. What is the probability of either \(X\) or \(Y\) occurring?

Step-by-Step Solution

Answer

The probability of either \( X \) or \( Y \) occurring is 0.70.

Step 1: Understanding the Problem

We need to find the probability of either event \( X \) or \( Y \) occurring given the individual probabilities \( P(X) = 0.55 \), \( P(Y) = 0.35 \), and the probability that both events occur \( P(X \cap Y) = 0.20 \).

Step 2: Applying the Addition Rule

To find the probability of either \( X \) or \( Y \) occurring, we use the formula: \[ P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \]This rule accounts for the overlap between the events, which is \( P(X \cap Y) \).

Step 3: Substitute the Given Values

Substitute the known values into the formula: \[ P(X \cup Y) = 0.55 + 0.35 - 0.20 \]

Step 4: Calculate the Probability

Perform the calculations: \[ 0.55 + 0.35 = 0.90 \]\[ 0.90 - 0.20 = 0.70 \]Thus, the probability of either \( X \) or \( Y \) occurring is 0.70.

Key Concepts

Addition Rule for Probabilities

The Addition Rule for Probabilities helps us find the probability that at least one of several events occurs. This is particularly useful when dealing with two overlapping events, like events X and Y in our example.
To use this rule properly, you must consider the entire overlap, which is when both events happen simultaneously.
The formula you use is:

• \( P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \)

The subtraction part \( P(X \cap Y) \) corrects for the double-counting of the intersection where both events happen.
This ensures that the total probability calculation is accurate and considers the overlapping possibilities rightfully.
Remember, events must be from the same sample space for the addition rule to work. It's always important to confirm if probabilities are independent or interdependent before applying.

Probability of Either Event

Often, you may seek to determine the likelihood of at least one of two events taking place instead of both or neither.
This is referred to as the "probability of either event" occurring. In mathematical terms, it is denoted by \( P(X \cup Y) \).
We can use the addition rule to find this probability, as seen in the previous section. It lets you compute not just the probability of each separate event, but also accounts for their simultaneous occurrence.
Using our example:

• The probability \( P(X) \) is 0.55, meaning event X is expected to occur 55% of the time.
• The probability \( P(Y) \) is 0.35, meaning event Y is expected to occur 35% of the time.

However, when both X and Y can occur together with \( P(X \cap Y) \) being 0.20, you should adjust the total probability by subtracting this overlap.
Ultimately, the probability of either X or Y, \( P(X \cup Y) \), is 0.70, indicating a 70% chance of one or both occurring.

Intersection of Events

The intersection of events refers to the scenario where two events occur simultaneously.
In probability terms, this intersection is denoted \( P(X \cap Y) \). The intersection is a crucial part of understanding dependent events and plays a significant role when using the Addition Rule for Probabilities.
For the example at hand, \( P(X \cap Y) \) is given as 0.20, meaning that both X and Y occur together 20% of the time.
Identifying this overlap allows us to ensure that our probability calculations for either event (or both) remain accurate and are not artificially inflated due to double-counting.
In practical tasks, evaluating intersections is pivotal for designing systems and predictions where two outcomes or phenomena occur together. This understanding assists in decision-making under uncertainty and making informed predictions.