4.1.2. Important Terms

Two events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$.

Independent events mean that the occurrence of one event does not affect the occurrence of the other. If events A and B are independent, knowing that A occurs does not change the probability of B occurring, and vice versa. Mathematically, this relationship is represented as the product of their individual probabilities: $P(A \cap B) = P(A) \cdot P(B)$. This indicates that the probability of both A and B occurring together is simply the product of their individual probabilities.

Think of flipping a coin and rolling a die at the same time. The outcome of the coin flip does not influence the outcome of the die roll. Therefore, these two events - getting heads on the coin (event A) and rolling a three (event B) - are independent.

Events that cannot occur at the same time. For mutually exclusive events A and B, $P(A \cap B) = 0$.

Mutually exclusive events are those that cannot happen simultaneously. If one event occurs, it precludes the occurrence of the other. In terms of probability, if you have two mutually exclusive events A and B, the probability that both A and B occur together is zero: $P(A \cap B) = 0$. This means if A happens, B cannot happen at all.

Imagine a scenario where you have a single dice. You cannot roll a number 1 and a number 2 at the same time; hence, rolling a 1 and rolling a 2 are mutually exclusive events.

A formula used to update probabilities based on new information.

Bayes' Theorem is a way to find the probability of an event based on prior knowledge of conditions related to the event. It allows us to update our beliefs in the light of new evidence. This theorem is mathematically expressed as $P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}$, where $P(A | B)$ is the conditional probability of A given B, $P(B | A)$ is the probability of B given A has occurred, and $P(B)$ is the probability of B.

Consider a medical test for a disease. Nearly everyone believes they have the disease based on a positive test result. However, with Bayes' Theorem, we can account for the accuracy of the test and the actual prevalence of the disease in the population to reassess how likely it is that a person actually has the disease after receiving a positive test result.

Useful for calculating unconditional probability using partitions.

The Total Probability Theorem connects conditional probabilities with marginal probabilities. It states that if you partition the entire sample space into disjoint events A1, A2, …, An, the probability of event B can be found by summing the conditional probabilities of B given each Ai, weighted by the probabilities of each Ai occurring. This can be expressed as $P(B) = \sum_{i} P(B|A_i) \cdot P(A_i)$.

Think of a basket of fruits. If you want to know the probability of picking an apple from the basket, you can break it down by categories like red apples, green apples, and other fruits. By finding the probabilities of picking an apple from each category (conditional probabilities) and then combining them based on their overall likelihood, you can determine your total probability of picking an apple.

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