Chapter 7: Problem 1

Compute the indicated quantity. \(P(B)=.5, P(A \cap B)=.2 .\) Find \(P(A \mid B)\)

### Short Answer

## Step by step solution

## Understand the known probabilities given in the problem.

## Use the formula for conditional probability to find the required probability.

## Substitute the given values in the formula and calculate the conditional probability.

## Interpret the result.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Probability of an Event

To conceptualize this, think of flipping a fair coin. The probability of getting heads (which is one possible event) is 0.5, or 50%. This means that out of a large number of coin flips, we expect about half of them to result in heads. Probabilities can be determined through experimentation, theoretical calculations, or established statistical methods.

When given a probability such as P(B) = 0.5 in our exercise, it informs us that there is an equal chance of event B occurring or not occurring in a single trial. Always remember, probabilities should not be interpreted as predictions for single events, but rather as the likelihood over many trials.

###### Intersection of Events

For example, if we have a deck of cards, the probability of drawing a card that is both red and a queen is the intersection of the two events: being red (Event A) and being a queen (Event B). To find this probability, one would need to know the total number of favorable outcomes (red queens) and the total number of possible outcomes (all cards).

In our textbook exercise, when we say P(A \(\text{\textcap}\) B) = 0.2, we're stating that the chance of both event A and event B happening together is 20%. This conceptual understanding of intersections is crucial when it comes to calculating conditional probabilities or understanding complicated events dependent on multiple outcomes.

###### Probabilistic Formulas

\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]

This formula states that the conditional probability of event A given event B, denoted as P(A | B), is the probability of the intersection of A and B (both events occurring) divided by the probability of B alone. In simple terms, it's asking: 'Given that B has occurred, what's the chance that A will also occur?'.

In our example, to find P(A | B), we used the known probabilities P(A \(\text{\textcap}\) B) = 0.2 (the intersection) and P(B) = 0.5 (the probability of event B occurring). According to the formula, the conditional probability is 0.4 or 40%. Such formulas are indispensable tools for statisticians and anyone interested in assessing probabilities in various scenarios, from simple games of chance to complex scientific experiments.