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The Law of Total Probability is a fundamental concept used to determine the overall likelihood of an event by considering all possible ways that event can occur. We do this by breaking down the event into scenarios, then weighing each scenario by its probability. This method gives a comprehensive view of the event's total probability.
In Nigel's case, the event in question is whether he manages to get to his morning classes. Since this can happen in two ways—either by setting his alarm or not setting it—the Law of Total Probability helps integrate these possibilities. Here's how:

• If the alarm is set, Nigel has a 90% chance of getting to class.
• If the alarm is not set, he has a 60% chance of still making it.

With these probabilities and knowing that the alarm is set 85% of the time, we apply the law:
\[ P(B) = P(B|A)P(A) + P(B| eg A)P( eg A) \]
By substituting the values we've:

• \(0.90 imes 0.85 = 0.765\) for when the alarm is set
• \(0.60 imes 0.15 = 0.09\) for when the alarm is not set

Adding these gives us 0.855, or 85.5%. This means Nigel gets to class on 85.5% of the days, combining both scenarios.

Bayes' Theorem is a powerful rule for determining the likelihood of an underlying cause when given the outcome. Essentially, it's about revising the probability of a condition based on new evidence. Think of it as flipping the conditional probability around. It’s especially useful for situations like Nigel’s, where we want to know the reverse probability of an event based on the result.
In part b of Nigel's situation, we know he made it to class and we seek to understand the likelihood he didn't set his alarm. Here’s how Bayes' Theorem fits in:

• We want \( P( eg A | B) \), the probability he didn't set the alarm given he arrived in class.
• We use Bayes' Theorem to calculate it: \[ P( eg A | B) = \frac{P(B | eg A)P( eg A)}{P(B)} \]

By inserting known values:

• \( P(B | eg A) = 0.60 \)
• \( P( eg A) = 0.15 \)
• \( P(B) = 0.855 \)

The resulting calculation shows that, after solving, the chance Nigel made it to class without setting the alarm is approximately 10.5%. Bayes' theorem provided a methodical way to understand this probability based on outcomes.

Conditional probability is all about figuring out the chance of an event happening, given that another event has already occurred. It's like saying, "Under these circumstances, how likely is this to happen?" This is a straightforward but deeply insightful part of probability theory.
In our scenario with Nigel, conditional probability helps us determine the chances of him going to class based on whether or not he sets his alarm. We use terms like \( P(B|A) \) and \( P(B| eg A) \), which stand for the probability of him getting to class given the alarm is set, and not set, respectively.

• If the alarm is set, then this probability is 90%. This means, knowing he's set the alarm, there's a high chance he makes it to class.
• If the alarm is not set, then this probability is 60%. This lowers since waking up without the alarm is more challenging.

These probabilities are conditional because they depend on the occurrence of a certain precursor event—setting the alarm or not. This framework helps detail the situation and calculate more sophisticated outcomes using the Law of Total Probability or Bayes' Theorem, making it foundational in grasping more complex probability concepts.