Question

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B},\) we have \(P(A)=0.4, P(B)=0.3,\) and \(P(A\) and \(B)=0.1.\) Find \(P(B\) if \(A).\)

Short answer

The probability of event B given that event A has occurred is 0.25.

Step-by-step solution

Identifying given values

From the problem, we've been given \( P(A) = 0.4 \), \( P(B) = 0.3 \), and \( P(A \cap B)= 0.1 \). The aim is to find the conditional probability \( P(B|A) \).

Apply the formula for conditional probability

The formula for the conditional probability is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \). Substituting the given values into the equation gives us \( P(B|A) = \frac{0.1}{0.4} \).

Evaluate the expression

Performing the division operation we get the final answer. This is the conditional probability of event B given that event A occurs.

Key concepts

Understanding P(A and B)

When we talk about the probability of two events happening together, we use the notation \(P(A \text{ and } B)\) or \(P(A \cap B)\). This represents the probability that both event A and event B occur. It's important to note that the events could be anything from the roll of a die, to drawing a card from a deck, or even more complex situations like predicting weather patterns.

To determine \(P(A \text{ and } B)\), we need to consider the nature of the events. Are they related, or do they affect each other's outcomes? Understanding this can help us decide whether to use straightforward multiplication of individual probabilities or a more nuanced approach. In contexts where events A and B are independent—meaning the occurrence of one does not affect the occurrence of the other—the calculation is simply the product of their individual probabilities: \(P(A) \times P(B)\).

However, if the events are not independent, as in many real-world situations, then their combined probability might be provided (as in the exercise), or we might need additional information to calculate it.

Probability of Event

The probability of an event is a measure of how likely that event is to occur. The event can range from something simple, like landing on heads when flipping a coin, to more complex events in fields like finance or meteorology. The probability is always between 0 and 1, where 0 means the event will not occur, and 1 means the event is certain to happen.

In everyday language, we might say an event is 'likely' or 'unlikely,' but in probability, we assign a numerical value to describe this. If the event is equally likely to occur or not occur, we assign a probability of 0.5, or 50%. As we saw in the textbook example, the probabilities were given: \(P(A)=0.4\) and \(P(B)=0.3\), implying that event A has a 40% chance of occurring, while event B has a 30% chance.

Probability Formula

A probability formula is used to calculate the likelihood of an event or a combination of events. The most basic formula is the ratio of the number of favorable outcomes to the total number of possible outcomes. This formula is often expressed as \(P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\).

For situations involving multiple events, such as finding \(P(A \text{ and } B)\), the formula may vary depending on whether events are independent or not. In the case of conditional probabilities like \(P(B|A)\), we use a specific formula: \(P(B|A) = \frac{P(A \cap B)}{P(A)}\). This formula, as applied in our exercise, allows us to find the probability of event B occurring given that event A has occurred, reflecting a sort of 'updating' of our belief about B after we know A has happened.

Statistical Independence

Two events are considered statistically independent if the occurrence of one does not influence the probability of the occurrence of the other. Suppose we flip a coin and roll a die simultaneously; the outcome of the coin toss doesn't affect the outcome of the die roll. They are independent events, and thus their combined probability is a simple product of their individual probabilities.

Why does independence matter? When dealing with multiple event probabilities, knowing whether events are independent allows us to quickly calculate combined probabilities without needing further information. On the other hand, if events are dependent, we cannot simply multiply the individual probabilities, and other methods, such as conditional probability, could come into play. In the exercise, the value of \(P(A \text{ and } B)\) is vital, but it does not tell us directly about the dependence of A and B; that's something we might infer or test with further analysis or additional data.