Problem 9 - Technical Mathematics with Calculus

Question

At a certain school, \(55 \%\) of the students have brown hair, \(15 \%\) have blue eyes, and \(7 \%\) have both brown hair and blue eyes. What is the probability that a student chosen at random will have either brown hair or blue eyes, or both brown hair and blue eyes?

Step-by-Step Solution

Answer

The probability that a student chosen at random will have either brown hair or blue eyes, or both is 0.63.

Step 1: Identify Relevant Probabilities

Identify the probability of having brown hair as P(Brown Hair) = 0.55, the probability of having blue eyes as P(Blue Eyes) = 0.15, and the probability of having both traits as P(Brown Hair and Blue Eyes) = 0.07.

Step 2: Apply the Inclusion-Exclusion Principle

Use the inclusion-exclusion principle to determine the probability of having either trait or both. The principle states that P(A or B) = P(A) + P(B) - P(A and B).

Step 3: Calculate the Probability

Substitute the given probabilities into the inclusion-exclusion formula: P(Brown Hair or Blue Eyes) = P(Brown Hair) + P(Blue Eyes) - P(Brown Hair and Blue Eyes) = 0.55 + 0.15 - 0.07.

Step 4: Simplify the Result

Simplify the expression to find the final probability: 0.55 + 0.15 - 0.07 = 0.63.

Key Concepts

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a critical concept in probability theory that allows us to calculate the probability of the union of two events by considering their individual probabilities and their intersection. In simpler terms, it helps us find the probability that at least one of several events will occur.

When dealing with two events, A and B, the inclusion-exclusion principle is expressed as:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
This formula accounts for the overlap between the events—we add the probabilities of each event occurring separately and then subtract the probability that they both occur since we counted that event twice. This principle can be extended to more than two events with additional terms to adjust for multiple overlaps.

Let's consider the practical implementation using the exercise above. We are given the probabilities of having brown hair \(P(Brown Hair) = 0.55\) and having blue eyes \(P(Blue Eyes) = 0.15\), along with the probability of both traits \(P(Brown Hair \text{ and } Blue Eyes) = 0.07\). Applying the inclusion-exclusion principle:
\[ P(Brown Hair \text{ or } Blue Eyes) = 0.55 + 0.15 - 0.07 \]
Through this principle, we avoid double-counting students with both traits and obtain the true probability of a randomly selected student having either brown hair, blue eyes, or both.

Probability Calculation

Probability calculation involves determining the likelihood of a particular outcome or set of outcomes. The foundation of probability is quite simple: it is typically expressed as a fraction or a decimal between 0 and 1, where 0 indicates impossibility and 1 signifies certainty. A probability of 0.5 represents just as likely to happen as not.

It is essential to correctly identify the various probabilities and their relationships when solving problems. In our example, the task is to calculate the probability of a student chosen at random having either brown hair or blue eyes, or both, given certain individual and joint probabilities. By using the inclusion-exclusion principle, we can calculate the combined probability while considering the intersection of those two characteristics.

After evaluating the given data and applying the formula, we simplify the result to obtain the final probability. Concluding from our example, the calculated probability of a randomly chosen student having brown hair or blue eyes is \(0.63\), which means there is a 63% chance of this combined event occurring.

Combinatorial Probability

Combinatorial probability is an area of probability that deals with counting and combining possibilities. It's pivotal when outcomes are discrete, and we want to count how many ways a particular event can happen. This topic often requires an understanding of permutations and combinations, which are methods used to count arrangements and selections without regard to order, respectively.

To calculate combinatorial probability, we determine the ratio of the favorable outcomes to the total possible outcomes. For instance, if you're trying to find out the probability of drawing a specific hand in a card game, you would count all the ways to make that hand (favorable outcomes) against all possible hands (total outcomes).

In our school example, combinatorial methods are less explicit but still underlying concepts. If we wanted to expand and ask how many students have either characteristic in a finite population, we'd use combinations to figure out the different ways these traits could be distributed among students. Understanding combinatorial probability underlines the importance of knowing how to count possible outcomes effectively, a key skill in the probabilistic toolkit.