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Conditional probability helps us find the likelihood of an event occurring based on the occurrence of another event. In our problem, we're interested in knowing the probability that Mary has a better hand, given that she decides to raise. This scenario involves calculating the probability of an event (Mary's hand being better) given another event (she raises) has happened already.

So, conditional probability is mathematically represented as:

• \( P(A | B) \): the probability of event \( A \) happening given \( B \) has already occurred.
• To compute this, we use the formula: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
• This approach helps update the probability of one event by considering the presence of another.

Without conditional probability, we wouldn't be able to accurately assess the situation based on new data, like in our poker hand problem.

Probability theory is the study of randomness and uncertainty. It's a branch of mathematics dealing with the likelihood that different events will occur. Fundamental to probability theory are experiments, which are actions or processes that yield uncertain outcomes.

In our poker scenario:

• We're examining the probability of Mary having a better hand than John.
• This is an example of a probabilistic event based on the information we are provided, like Mary's decision to raise.

Probability theory gives us the tools to quantify and analyze the chances of events happening. This is crucial when making decisions based on incomplete information, such as determining the likelihood of Mary's hand being better based on her betting behavior.

In probability, an 'event' is any outcome or set of outcomes from a random process. In our example, specific events are being considered to address the problem.

Here's how event definition comes into play:

• Event \( B \) is defined as "Mary has a better hand than John".
• Event \( R \) is defined as "Mary raises".

Identifying and clearly defining events is crucial for setting up any probability problem efficiently. It allows us to use probability laws and theorems to compute the likelihood of complex outcomes, as we do in our poker hand problem. Without clear event definitions, applying probabilistic concepts becomes challenging.

The total probability theorem combines probabilities of different disjoint events to determine the likelihood of an overall event. It helps in calculating the overall probability of an event when it can occur from various independent scenarios.

In our example, we use the total probability theorem to find the probability that Mary raises, which involves combining the chances in two different cases:

• Mary raises with a better hand.
• Mary raises with a poorer hand.

Mathematically, this is expressed as:\[ P(R) = P(R \mid B)P(B) + P(R \mid B^c)P(B^c) \]This formula allows us to find \( P(R) \) by considering both scenarios, which is essential before applying Bayes' Theorem to find the conditional probability. Understanding this theorem is key for handling complex probability scenarios that depend on multiple conditions or events.