Question

In Exercises \(\mathrm{P} .8\) to \(\mathrm{P} .14\), use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Find \(P(\operatorname{not} A).\)

Short answer

The probability of 'not A' is 0.2

Step-by-step solution

Identify the Given

From the exercise, we know that the probability of event A, denoted as \(P(A)\), is 0.8.

Apply the formula

The probability of 'not A' can be found by subtracting the probability of A from 1. This gives us \(P(\operatorname{not} A) = 1- P(A)\). Substituting the given \(P(A)\) value into this equation gives us \(P(\operatorname{not}A) = 1 - 0.8\).

Calculate the Probability of 'not A'

Finally, performing the subtraction gives us \(P(\operatorname{not} A) = 1 - 0.8 = 0.2\). So, the probability of 'not A' is 0.2.

Key concepts

Complementary Events

When dealing with probabilities, it's crucial to understand the concept of complementary events. These are a pair of outcomes of an event that are the only two possible outcomes and that cannot occur simultaneously. In more straightforward terms, if one event happens, the other cannot, and vice versa. For any event A, the complement is 'not A', often written as \( A' \).

Imagine flipping a coin. The complementary events in this scenario would be the coin landing on heads (\( A \) – the event that the coin shows heads) and the coin not landing on heads (\( A' \)). Since the coin has to land on one side, if it doesn't land on heads, it must land on tails. Thus, \( P(A) + P(A') = 1 \), where \( P \) stands for the probability of an event occurring. Understanding this relationship is fundamental for solving many probability problems and is a foundational concept in statistics.

Subtracting Probabilities

Subtracting probabilities is a technique often used in finding the probability of complementary events or the difference between the probabilities of two events occurring. As seen in the exercise provided, to calculate \( P(\text{not } A) \), we subtracted \( P(A) \) from 1. This is because the sum of the probabilities of an event and its complementary event always equals 1.

Here's why this makes sense: Since one of the two complementary events must happen, their combined probabilities cover all possible outcomes, which is sure to happen (hence, a total probability of 1). So, by subtracting the probability of the known event from 1, we confidently determine the probability of the complementary event. It's like saying, 'If it's not this, it's definitely the other,' because there are no other options. This idea is instrumental in various probability scenarios, such as deducing unknown probabilities when only their complementary probabilities are known.

Probability Formulas

Probability formulas provide us with the mathematical language needed to calculate and express the likelihood of different events. The most basic formula is that the probability of any event A is the number of ways event A can occur divided by the total number of possible outcomes. This ratio is often expressed as \( P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \).

Other important probability formulas include the Addition Rule, useful when determining the probability that at least one of two events occurs, and the Multiplication Rule, which calculates the probability of two independent events occurring together. To elucidate further with an example from the exercise, the probability of an event not occurring, also called the complement rule, is \( P(A') = 1 - P(A) \).

Remembering and correctly applying these formulas is pivotal for solving probability exercises effectively. Whether you're computing the odds of a single, isolated event or dealing with complex scenarios involving multiple independent or dependent events, knowing the right formulas is key to unlocking the answers.