4.1.3. Formulae Summary

Concept Formula
Conditional Probability 𝑃(𝐴|𝐵) = \frac{P(A \cap B)}{P(B)}

The formula for conditional probability expresses how we can calculate the likelihood of event A occurring given that event B has already happened. In this formula, $P(A \cap B)$ represents the probability that both A and B occur at the same time. The denominator, $P(B)$, is the probability that event B occurs. We use this relationship because we're only interested in the situation where B has happened, thus limiting our sample space.

Imagine you are choosing a card from a deck. If we know the card drawn is a heart, the probability of drawing an ace given that information becomes clearer since we're only considering the hearts, not the entire deck. There are 13 hearts, and only one of them is an ace, hence the conditional probability is much more focused.

Concept Formula
Product Rule 𝑃(𝐴 ∩ 𝐵) = P(A) \cdot P(B|A)

The Product Rule allows us to compute the joint probability of events A and B occurring together by taking the probability of A occurring and multiplying it by the conditional probability of B given A. This is helpful in situations where the occurrence of one event affects the chance of the other event's occurrence.

Think of it like making a sandwich. The chance of you having all the ingredients (bread, lettuce, etc.) is the first probability. Then, given you have the bread, the chance of putting lettuce on it becomes the second conditional probability. Multiplying these probabilities gives the likelihood of making that specific sandwich.

Concept Formula
Bayes’ Theorem 𝑃(𝐴|𝐵) = \frac{P(B|A) \cdot P(A)}{P(B)}

Bayes' Theorem provides a way to update our beliefs or probabilities based on new evidence. It relates the conditional and marginal probabilities of random events. Here, it allows us to find the probability of A occurring given that B has occurred by considering the probability of B occurring if A is true, adjusted for our prior belief in A.

Consider a doctor diagnosing a disease based on test results. If a patient has a positive test result (event B), Bayes' theorem helps the doctor update the probability of the patient actually having the disease (event A) taking into account how accurate the test is and the general prevalence of the disease.

Concept Formula
Total Probability P(B) = \sum_{i} P(B|A_i) \cdot P(A_i) for mutually exclusive and exhaustive events A_i

The Total Probability Theorem helps calculate the overall probability of event B by considering all possible ways that B can occur through different scenarios or events A_i. Each term factors in the probability of each scenario and how likely B is within that scenario, ensuring that we account for all possibilities exhaustively.

Imagine you want to know the likelihood of rain tomorrow. You could have different weather systems affecting whether it rains (sunny, cloudy, stormy). By calculating the chance of rain for each of those weather patterns and how likely each pattern is to happen, you can determine the overall probability of rain tomorrow.

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