Question

Refer to Exercise 6.18. Adding probabilities in the first row of the given table yields \(P(\) midsize \()=.45\), whereas from the first column, \(\mathrm{P}\left(4 \frac{3}{8}\right.\) in. grip) \(=.30\). Is the following true? $$ P\left(\text { midsize } \text { or } 4 \frac{3}{8} \text { in. grip }\right)=.45+.30=.75 $$ Explain.

Short answer

Without further information on the relationship between these events, it cannot be definitively stated whether \(P(\text{midsize or } 4 \frac{3}{8} \text{ in. grip}) = .45 + .30 = .75\). The question hinges on whether the events are mutually exclusive or not.

Step-by-step solution

Understanding Mutually Exclusive Events

Two events are said to be mutually exclusive if they cannot both occur at the same time. In this case, if 'midsize' and '4 3/8 inch grip' are mutually exclusive, the probability of either event occurring would indeed be the sum of their individual probabilities.

Considering Overlapping Events

If the events can occur simultaneously, they are not mutually exclusive and we risk 'double counting' when we simply add the probabilities. In such a case, the probability of either event occurring is equal to the sum of individual probabilities minus the probability of both events occurring together.

Conclusion

To determine whether \(P(\text{midsize or } 4 \frac{3}{8} \text{ in. grip}) = .45 + .30 = .75\) is true, we need additional information about whether these events can occur at the same time. Without this information, we cannot definitively say if the provided equation is accurate.

Key concepts

Mutually Exclusive Events

In probability theory, understanding mutually exclusive events is crucial. These are events that cannot occur at the same time. For example, when flipping a coin, the result can only be heads or tails, not both—these outcomes are mutually exclusive.

Mathematically, if we have two mutually exclusive events, A and B, their combined probability is the sum of their individual probabilities: \( P(A \text{ or } B) = P(A) + P(B) \). This is known as the Addition Rule for Mutually Exclusive Events.

In the problem presented, the term 'midsize' and the specification '4 \frac{3}{8} inch grip' would be considered mutually exclusive if a tennis racket cannot be both simultaneously. If that's the case, then logically, the probability that a racket is either midsize or has a 4 \frac{3}{8} inch grip would indeed be the sum of their individual probabilities, 0.45 and 0.30, respectively.

Overlapping Events

Unlike mutually exclusive events, overlapping events can occur at the same time. These might be two characteristics that are not mutually exclusive, like a person who is both a teacher and a parent.

When dealing with overlapping events, the addition of their probabilities must account for their intersection to avoid double-counting. The correct formula in such cases is: \( P(A \text{ or } B) = P(A) + P(B) - P(A \cap B) \), where \( P(A \cap B) \) is the probability of both A and B occurring together.

In the context of our textbook problem, if the 'midsize' tennis rackets can sometimes have a '4 \frac{3}{8} inch grip', then the events overlap. The probability equation in question would not be correct unless we subtract the probability that a racket is both midsize and has a 4 \frac{3}{8} inch grip. Without that piece of information, we cannot ascertain the true probability of getting either a midsize racket or one with a 4 \frac{3}{8} inch grip.

Probability Theory

Probability theory is a mathematics branch that deals with the likelihood of different events occurring. It is foundational for various fields such as statistics, finance, and risk assessment.

The fundamental principle of probability is that the sum of probabilities for all possible outcomes in a space is 1. When it comes to events, probabilities range between 0 and 1, with 0 indicating an impossible event and 1 a certain one.

Understanding events as mutually exclusive or overlapping is essential in applying probability theory correctly. Students must recognize the type of events they are working with in problems, as this determines the mathematical approach to finding probabilities. The application of the correct formula, whether for mutually exclusive events or overlapping ones, leads to accurate solutions and greater comprehension of the complex scenarios that probability theory often deals with.

Essentially, good grasp of probability theory helps students not just in mathematics, but in making informed decisions based on likelihoods in everyday life.