Problem 12 - Basic Statistics for Business and Economics with Student

Question

The events \(X\) and \(Y\) are mutually exclusive. Suppose \(P(X)=.05\) and \(P(Y)=.02 .\) What is the probability of either \(X\) or \(Y\) occurring? What is the probability that neither \(X\) nor \(Y\) will happen?

Step-by-Step Solution

Answer

P(X ∪ Y) = 0.07; P(neither X nor Y) = 0.93.

Step 1: Understand Mutually Exclusive Events

Events are mutually exclusive when they cannot occur at the same time. For events X and Y, mutually exclusive means if one happens, the other cannot. Therefore, the probability of both X and Y happening together is 0.

Step 2: Calculate the Probability of Either X or Y Occurring

Since X and Y are mutually exclusive, the probability of either X or Y occurring is the sum of their probabilities. This is calculated as follows:\[P(X \cup Y) = P(X) + P(Y) = 0.05 + 0.02 = 0.07.\]

Step 3: Calculate the Probability That Neither X Nor Y Happens

The probability that neither event X nor event Y occurs is the complement of the probability of either X or Y occurring. The formula for the complement is:\[P(\text{neither X nor Y}) = 1 - P(X \cup Y) = 1 - 0.07 = 0.93.\]

Step 4: Conclusion

The probabilities have been calculated. The probability of either event X or event Y occurring is 0.07, and the probability that neither event X nor event Y occurs is 0.93.

Key Concepts

Probability Calculations

Probability calculations are a fundamental part of statistical analysis and help us predict how likely various outcomes are. When we talk about calculating probability, we're essentially trying to find out how often we can expect a particular event to occur. Here are a few key points to consider:

• Probability is typically expressed as a number between 0 and 1, where 0 means an event will certainly not occur, and 1 means it will certainly occur.
• The probability of an event is calculated by dividing the number of ways that event can happen by the total number of possible outcomes.
• For example, if you roll a six-sided die, the probability of rolling a three is given by a straightforward calculation: there is 1 way to roll a three and 6 possible outcomes, giving a probability of \(\frac{1}{6}\).

These calculations are crucial for understanding different scenarios, whether in gambling, weather forecasts, or even just deciding on everyday matters.

Complement Rule

The complement rule in probability is a handy way to find out how likely something is not to happen. This concept simplifies many problems by focusing on what doesn't happen instead of what does.

• The complement of an event A is everything in the sample space that is not A.
• Using the complement rule, we can calculate the probability of the complement of event A as \(P(A') = 1 - P(A)\).
• This is especially useful when calculating the probability of something not happening is more straightforward than the event itself. For example, knowing it won't rain might be easier to determine than knowing it will rain.

This rule is not only a time-saver but often makes our calculations more straightforward by dealing with probabilities that add up to a whole.

Probability Theory

Probability theory forms the backbone of our understanding of chance and uncertainty. This branch of mathematics provides a set of tools to explain how likely events are to happen.Its key pillars are:

• Mutual Exclusivity: Events that cannot happen at the same time. For instance, when flipping a coin, getting a heads and a tails at the same time is impossible, making these mutually exclusive events.
• Addition Rule: For mutually exclusive events, the probability of either occurring is simply the sum of their probabilities. As seen in the exercise, this is why \(P(X \cup Y) = P(X) + P(Y)\).
• Independence: Events that don't affect each other's outcomes. This is not the case with mutually exclusive events but is another critical component of probability theory.

This theory allows us to model and predict real-life situations, making it an invaluable part of decision-making and science.