Question

Use the fact that we have independent events \(\mathrm{A}\) and \(\mathrm{B}\) with \(P(A)=0.7\) and \(P(B)=0.6\). Find \(P(A\) and \(B)\).

Short answer

The probability of both events A and B occurring, \(P(A \text{ and } B)\) is 0.42

Step-by-step solution

Understand the given probabilities

We are given that the probability of event A, denoted by \(P(A)\), is 0.7 and the probability of B, denoted by \(P(B)\), is 0.6.

Confirm that the events are independent

The problem states that events A and B are independent. Thus, the occurrence or nonoccurrence of event A doesn't affect the probability of event B, and vice versa.

Apply the rule for independent events

The rule for independent events states that the probability of both events A and B happening, denoted by \(P(A \text{ and } B)\), is equal to the probability of event A multiplied by the probability of event B. We write this as \(P(A \text{ and } B) = P(A)*P(B)\)

Calculate \(P(A \text{ and } B)\)

By substituting our known probabilities into the equation from step 3, we get \(P(A \text{ and } B) = 0.7 * 0.6 = 0.42\)

Key concepts

Probability Theory

Probability theory lies at the heart of many scientific disciplines, offering a formal framework to predict the likelihood of events. It is a branch of mathematics that deals with calculating the chances of a specific outcome. In essence, it helps us quantify uncertainty. Understanding probability can assist in making informed decisions based on the likelihood of various outcomes.

For instance, suppose a weather forecast indicates a 70% chance of rain tomorrow (\(P(\text{{Rain}}) = 0.7\)). This percentage reflects the probability theory in action, providing a measure of how likely it is to rain. The theory accommodates outcomes ranging from the most certain (with probability 1 or 100%) to the impossible (with probability 0 or 0%).

Within the study of probability, events can be classified in multiple ways, such as independent events, which do not influence each other, and dependent events, where the outcome of one can affect the outcome of another. This classification is crucial for applying the correct rules when calculating probabilities.

Independent Events Rule

The independent events rule is a critical concept within probability theory, particularly when considering the relationship between two or more events. Two events are said to be independent if the occurrence of one event does not affect the occurrence of the other. In other words, the outcome of one event gives no information about the other.

For example, when flipping a coin and rolling a die, the result of the coin toss (heads or tails) has no impact on what number the die will land on. This is a classic case of independence between two events. Recognizing when events are independent is essential because it allows us to apply a simplified method of calculating the combined probability of these events occurring. If we improperly assume independence, our probability calculations could lead to incorrect conclusions.

Independent events often appear in various real-world scenarios, such as the probability of drawing an ace from a deck of cards and then rolling a six on a die, where both events do not interfere with each other's outcomes.

Probability Multiplication Rule

The probability multiplication rule is a fundamental concept when dealing with independent events. It states that the probability of two independent events occurring together (concurrently) is equal to the product of their separate probabilities.

To apply this rule, we use the formula: \
\[P(A \text{ and } B) = P(A) \times P(B)\]
where \(P(A)\) and \(P(B)\) are the probabilities of events A and B happening individually. Returning to our textbook example, since events A and B are independent with probabilities 0.7 and 0.6 respectively, we simply multiply these probabilities to get \(P(A \text{ and } B) = 0.7 \times 0.6 = 0.42\).

This approach allows for a streamlined calculation when dealing with independent events, avoiding more complex probability methods needed for dependent events. It's a powerful tool for predicting the likelihood of multiple events occurring in tandem, and is widely used across different fields such as statistics, gambling, finance, and risk assessment.