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The Law of Total Probability is a fundamental concept in probability theory that allows us to determine the total probability of a specific event based on the probabilities of that event occurring under several different circumstances. In simple terms, it helps us piece together the overall probability from its parts.
To use this law, we first need to identify all the possible ways an event can occur and then calculate the probability for each of these ways. Each part's probability is multiplied by the chance of that scenario happening and then summed together. This total gives us a comprehensive view of how likely the event is to happen.
In the context of our exercise, we used the Law of Total Probability to find the probability that a passenger is found with a weapon, identified as \( P(\text{Weapon}) \). We combined the likelihood of a weapon being detected at each airport, weighted by the proportion of total traffic handled by each airport:

• For Airport A: \( P(\text{Weapon }| A) \times P(A) \)
• For Airport B: \( P(\text{Weapon }| B) \times P(B) \)
• For Airport C: \( P(\text{Weapon }| C) \times P(C) \)

Putting this all together, we find that \( P(\text{Weapon}) = 0.86 \).

Conditional Probability is key to understanding how one event affects the likelihood of another, providing a filter that refines our probability predictions based on new information. It's like updating your guess when you know something specific has already happened.
More formally, conditional probability is denoted as \( P(A|B) \), representing the probability of event A occurring given that event B has already occurred.
In our exercise, we wanted to find \( P(A|\text{Weapon}) \) and \( P(C|\text{Weapon}) \). This translates to knowing a weapon was detected, and wanting to know the likelihood it happened at Airport A or C, respectively. By applying Bayes' Theorem, which cleverly leverages conditional probabilities, we could reverse the cause-and-effect scenario to find:

• \( P(A|\text{Weapon}) = \frac{P(\text{Weapon}|A) \times P(A)}{P(\text{Weapon})} \)
• \( P(C|\text{Weapon}) = \frac{P(\text{Weapon}|C) \times P(C)}{P(\text{Weapon})} \)

After plugging in the known values, we deduced \( P(A|\text{Weapon}) \approx 0.523 \) and \( P(C|\text{Weapon}) \approx 0.197 \), indicating the probability that a found weapon originated at Airports A and C respectively.

Detection Rates are particular probabilities that quantify the effectiveness of security measures at identifying potential threats. In this context, these rates reflect how proficient each airport is in detecting weapons that passengers try to bring through.
Detection rates are crucial because they impact the reliability of our overall probability calculations. They directly affect the conditional probabilities used in Bayes' Theorem, indicating how likely it is for a passenger to be caught with a weapon at a given airport.
In the exercise, detection rates were specific for each airport:

• Airport A: 0.9, meaning a 90% chance of successfully detecting a weapon.
• Airport B: 0.8, meaning an 80% chance.
• Airport C: 0.85, meaning an 85% chance.

These rates provided the \( P(\text{Weapon}|A) \), \( P(\text{Weapon}|B) \), and \( P(\text{Weapon}|C) \) values needed for both the Law of Total Probability and Bayes' Theorem, demonstrating their pivotal role in assessing security performance and the likelihood of different scenarios.