Question

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 \(P(A\) or \(B)\).

Short answer

So, the probability of either A or B occurring, \(P(A \text{ or } B)\), is 0.95.

Step-by-step solution

Identify the given probabilities

The probability of event A is given as \(P(A)=0.8\), the probability of event B is \(P(B)=0.4\), and the probability of the intersection of A and B is \(P(A \text{ and }B)=0.25\).

Apply formula for probability of union of two events

The formula for the probability of A or B is given by \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\). Now, substitute the given probabilities into this formula.

Calculate

Using the given values and substituting them into the formula, the calculation will look like this: \(P(A \text{ or } B) = 0.8 + 0.4 - 0.25 = 0.95\).

Key concepts

Understanding Probability

When we talk about probability, we're referring to the likelihood of a particular event occurring. It is a numerical measure that ranges from 0 to 1, where 0 indicates an impossible event and 1 represents a certainty. In real-world scenarios, probabilities are essential as they help us to assess risks and make informed decisions.

For example, if we toss a fair coin, we know that the chance of getting 'heads' is as likely as getting 'tails'. Therefore, we say the probability of getting a head (\( P(\text{Head}) \) ) is 0.5. If we dive deeper, in probability there are various kinds of events - for instance, 'simple events' which can't be broken down further, and 'compound events' which are outcomes of two or more simple events.

Intersection of Events

The intersection of events is a fundamental concept in probability theory. It refers to the occurrence of two or more events at the same time. When we say 'A and B', we are looking for the probability of both events A and B happening together. This is denoted as \( P(A \text{ and } B) \) or \( P(A \text{ and I expected behavior or resultI expected behavior or result} B) \), depending on the notation preference.

For instance, if you have a deck of cards, drawing a 'red card that is also a king' would represent the intersection of two events: drawing a red card (\( A \)) and drawing a king (\( B \)). Probability of intersection plays a crucial role in understanding how different events relate to each other and is used to calculate more complex probabilities.

Formula of Probability

The formula for probability provides a clear mathematical way to calculate the likelihood of different types of events. A key formula in this area is the one used to find the probability of the union of two events, which describes the likelihood of either event A or event B occurring. The formula is \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \).

The formula ensures that we are not counting the intersection of A and B twice. This is essential because, in the context of probability, each possible outcome should be considered once and only once for accurate calculation. Accurate application of this formula allows us to solve many probability problems efficiently, combining our knowledge of individual event probabilities and how they intersect.