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The Law of Total Probability is a fundamental rule that helps us relate conditional probabilities with marginal probabilities. It provides a way to express the probability of an event, based on a partition of the sample space. Let's break this down simply:

If we have a set of mutually exclusive events \(A_1, A_2, \ldots, A_n\) that cover the entire sample space and an event \(B\), the probability of \(B\) can be expressed as:

• \(P(B) = P(B \cap A_1) + P(B \cap A_2) + \ldots + P(B \cap A_n)\)

A more useful form is using conditional probabilities, which states:

• \(P(B) = P(B \mid A_1)P(A_1) + P(B \mid A_2)P(A_2) + \ldots + P(B \mid A_n)P(A_n)\)

In the context of the given problem, we use the concept of conditional probabilities like \(P(B \mid A)\) and \(P(B' \mid A)\) to establish comparisons. Here, the law helps in connecting these probabilities by using the fact that their sum is always one: \(P(B \mid A) + P(B' \mid A) = 1\).

This understanding is key in analyzing how observing event \(A\) effects both \(B\) and its complement \(B'\).

Complementary events are pairs of events where one event occurs if and only if the other does not. Mathematically, if \(B\) is an event, then its complement \(B'\) occurs whenever \(B\) does not. This means:

• \(P(B) + P(B') = 1\)

In terms of probability, this relationship ensures that there is a balance. The total probability across the outcomes covered by both \(B\) and \(B'\) adds up to certainty (1).

In the problem you are working on, this principle is crucial to understand the inequality \(P(B' \mid A) < P(B')\). Knowing that the probability of an event and its complement sums to one helps reveal changes in the probability landscape due to additional information, like knowing event \(A\) has occurred.

This underpins the step where we rewrite \(1 - P(B)\) as \(P(B')\) and translate that into the inequality involving conditional probabilities. The complement's altered condition an observation must inform is how likelihoods adjust when prefacing information about \(A\) is engaged.

In the realm of probability, inequalities often arise when comparing conditional probability against unconditional probability. These can reveal relationships and insights about how events relate under different conditions.

In this specific problem, the inequality \(P(B \mid A) > P(B)\) indicates that knowing \(A\) has occurred increases the likelihood of event \(B\). This sets a context: event \(A\) positively correlates with \(B\). When using the complement \(B'\), the original inequality transforms to show that the chance of \(B'\) is actually reduced when \(A\) occurs, as summarized in \(P(B' \mid A) < P(B')\).

• This highlights that the condition set by \(A\) skews probability favorably to \(B\)'s realization over its absence.
• It helps conceptualize the events' dynamics, showcasing interdependencies and impacts conditional information can assert.

The inequality presents a powerful way to understand shifts in probability landscapes when additional conditions inform event likelihoods. By evaluating inequalities, one gains insight into how new information rearranges the probabilities of events occurring.