91Ó°ÊÓ

In probability theory, when we say that two events, like \(A\) and \(B\), are equally likely, it means that the probability of each event occurring is the same. In mathematical terms, this is expressed as \(P(A) = P(B)\).
For example, when flipping a fair coin, getting "heads" is equally likely as getting "tails," both having a probability of \(0.5\).
Equally likely events allow us to simplify complex probability problems by assuming the same likelihood for each event.
In our exercise, as \(A\) and \(B\) are equally likely, we use a common symbol, \(p\), for their probabilities: \(P(A) = p\) and \(P(B) = p\).
This simplification helps streamline the equations when determining the intersection or union of such events.

The intersection of events in probability refers to the situation where two or more events happen at the same time.
Mathematically, for two events \(A\) and \(B\), their intersection is defined as \(P(A \cap B)\). It represents the probability that both \(A\) and \(B\) occur simultaneously.
When dealing with equally likely events and assuming independence, the probability of their intersection can be found by multiplying their individual probabilities: \(P(A \cap B) = P(A) \times P(B)\).
In our example, since \(A\) and \(B\) are equally likely, we use \(p\) for both, resulting in \(P(A \cap B) = p^2\).
Understanding intersection is crucial for solving problems where you need to find overlapping probabilities, especially when constraints, like a minimum value, are placed on the intersection.

Inequality in probability requires us to find probabilities that satisfy a specific condition or constraint. In the given exercise, we have the inequality \(p^2 \geq 0.98\), reflecting a condition on the intersection of \(A\) and \(B\).
Solving this requires mathematical manipulation, often involving algebraic or calculus-based techniques, depending on the problem's complexity.
Here, to solve \(p^2 \geq 0.98\), take the square root of both sides. This yields \(p \geq \sqrt{0.98}\), simplifying the inequality to determine the acceptable range for \(p\).
Inequalities are common in probability, helping define constraints that ensure the validity of scenarios under specific conditions and guiding towards informed conclusions.