Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(E\) and \(F\) be independent events; show that \(E\) and \(F^{c}\) are independent.

Step-by-Step Solution

Answer

To show that events E and \(F^c\) are independent, we need to prove that \(P(E \cap F^c) = P(E) * P(F^c)\). First, we express \(P(E \cap F^c)\) as \(P(E) - P(E \cap F)\). Since E and F are independent, we know that \(P(E \cap F) = P(E) * P(F)\). Substituting this expression, we get \(P(E \cap F^c) = P(E) - P(E) * P(F)\). By rewriting the right-hand side of the equation, we obtain \(P(E \cap F^c) = P(E) * (1 - P(F))\), which is the same as \(P(E) * P(F^c)\). Therefore, E and \(F^c\) are independent events.

Step 1: Express P(E ∩ F^c) in terms of E and F and their intersection

Let us start by writing down the probability of event E ∩ F^c: \(P(E \cap F^c) = P(E - (E \cap F))\) Now let's recall the general formula for calculating the probability of the difference of two events: \(P(A - B) = P(A) - P(A \cap B)\) Using this formula, we rewrite the probability of event E ∩ F^c in terms of E, F, and their intersection: \(P(E \cap F^c) = P(E) - P(E \cap F)\)

Step 2: Use the independence of E and F to simplify the expression

Since E and F are given as independent events, we know that: \(P(E \cap F) = P(E) * P(F)\) Now we can substitute this expression into our original equation: \(P(E \cap F^c) = P(E) - P(E) * P(F)\)

Step 3: Show that P(E ∩ F^c) = P(E) * P(F^c)

Finally, we rewrite the right-hand side of the equation to express it as the product of P(E) and P(F^c): \(P(E \cap F^c) = P(E) * (1 - P(F))\) Since the complement of an event F is given by \(F^c = 1 - F\), we can replace (1 - P(F)) with P(F^c): \(P(E \cap F^c) = P(E) * P(F^c)\) This completes the proof. Since \(P(E \cap F^c) = P(E) * P(F^c)\), the events E and F^c are independent.

Key Concepts

Probability of Independent Events

Understanding the probability of independent events is essential when solving problems where two or more events do not influence each other. Independent events are those where the occurrence of one event does not affect the probability of the other event occurring. For example, flipping a coin and rolling a die are independent events because the outcome of the coin toss doesn't change the chances of rolling a particular number on the die.

To calculate the probability of both independent events occurring, you multiply the probability of one event by the probability of the other. The mathematical formula is simply: \[ P(E \cap F) = P(E) \times P(F) \]
Using this concept helps us understand complex probability scenarios by breaking them down into simpler, unrelated events.

Complement of an Event

The complement of an event, denoted as 'event complement,' is another essential concept which represents all outcomes that are not part of the original event. If you have an event 'A', the complement 'A^c' is everything that 'A' is not. For understanding probability, it's crucial to see the whole picture, which includes what could happen outside the desired event.

Mathematically, the probability of an event's complement is: \[ P(A^c) = 1 - P(A) \]
This equation expresses that the probability of not occurring (event complement) and the probability of occurring (event itself) always add up to 1, the certainty of all possible outcomes. This idea is instrumental when considering events and their non-occurrences together.

Intersection of Events

The intersection of events comes into play when you're looking for the combined occurrence of two events. When you see the symbol '\(\cap\)', it refers to the intersection, meaning 'and' in probability. It asks the question: What is the likelihood that both events happen simultaneously?

In practice, to find the probability of the intersection of two events 'E' and 'F', we usually multiply the probability of 'E' by the probability of 'F', but only if they are independent, as discussed earlier. For dependent events (those that affect each other), we must calculate this differently because the outcome of one event affects the outcome of the other.

Probability Formulas

Understanding a variety of probability formulas can arm students with the tools necessary to tackle various probability problems. Two essential formulas that have been discussed are:

• The multiplication rule for independent events: \( P(E \cap F) = P(E) \times P(F) \)
• The complement rule: \( P(A^c) = 1 - P(A) \)

There are additional formulas, such as the addition rule for mutually exclusive events, Bayes' theorem for conditional probability, and the formula for total probability, among others. These formulas are foundational to solving probability problems, and recognizing when and how to apply them is a vital skill in probability and statistics.