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 \(B)=0.1.\) Are events \(A\) and \(B\) independent?

Short answer

Events \(A\) and \(B\) are not independent because \(P(A) \cdot P(B) \neq P(A \text{ and } B)\).

Step-by-step solution

Identify given information

The probabilities of events \(A\) and \(B\) are given: \(P(A) = 0.4\), \(P(B) = 0.3\), and the probability of the intersection of \(A\) and \(B\) is given: \(P(A \text{ and } B) = 0.1\).

Calculate the product of individual probabilities

Calculate the product of probabilities of \(A\) and \(B\): \(P(A) \cdot P(B) = 0.4 \cdot 0.3 = 0.12\).

Compare individual probabilities product with intersection probability

We compare the calculated value in step 2 with the given intersection probability \(P(A \text{ and } B)\). If \(P(A) \cdot P(B) = P(A \text{ and } B)\), then events \(A\) and \(B\) are independent.

Key concepts

Event Independence

Understanding event independence is a critical step in mastering probability theory. Two events, let's call them A and B, are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event. In mathematical terms, this means that if events A and B are independent, then the probability of both events occurring together, which is the intersection denoted as P(A and B), is equal to the product of their individual probabilities, P(A) * P(B).

To put this into context, let's consider a practical example: If you flip a coin and roll a die at the same time, the probability of getting heads on the coin does not influence the probability of rolling a six on the die. These events are independent. However, if you were drawing two cards successively from a deck without replacement, the outcome of the first draw would affect the second, making them dependent events.

Intersection of Events

The concept of the intersection of events plays a vital role in probability. When we talk about the intersection of two events, A and B, denoted as P(A and B), we are referring to the probability that both events occur together. It's the overlap between two sets of outcomes.

For example, if event A is 'it rains on Monday' and event B is 'it is windy on Monday,' the intersection of these two events is 'it is rainy and windy on Monday.' The intersection becomes fundamental when we consider whether events are independent or not.

Visualizing Intersection

A common way to visualize intersections in probability is by using a Venn diagram. Two overlapping circles can represent events A and B, with the overlap area representing the intersection where both A and B occur.

Probability Calculation

Probability calculation is the quantitative assessment of how likely an event is to occur. The fundamental principle of probability can be stated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Probabilities are always expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 represents a certain event.

Calculating the probability of independent events involves multiplying their individual probabilities. For instance, if you're trying to find the probability of drawing a red card from a deck and then flipping heads on a coin, you multiply the probability of drawing a red card (1/2) by the probability of flipping heads (1/2), resulting in a combined probability of 1/4, assuming the events are independent.