Probability

Mutually exclusive events: We check if two events A and B are mutually exclusive if they verify:

Independent events: We check if two events A and B are independent if they verify:

If two events are mutually exclusive (they cannot occur at the same time), they CANNOT be independent.

Law of total probability: Let $(A_1, A_2, ..., A_n)$ be pairwise mutually exclusive events whose union is the sample space (E), then, the probability of any other event B is given by:

Example: Out of 60 high school students, 25 are freshmen and 35 are sophomores. 60% of the freshmen pass physics, and 75% of the sophomores do. A student is chosen at random. What is the probability that they pass physics?
Solution: Let the events be:

• $\mathbf{A_1}$: freshmen students, with $P(A_1) = 25/60$
• $\mathbf{A_2}$: sophomore students, with $P(A_2) = 35/60$
• $\mathbf{B}$: students who pass physics, and $\mathbf{P(B)}$ is what is asked:

where
$P(B/A_1)$ Probability of passing physics if they are a freshman $(0.6)$
$P(B/A_2)$ Probability of passing physics if they are a sophomore $(0.75)$. Then:

Bayes' Theorem, posterior probability: Let $(A_1, A_2, ..., A_n)$ be pairwise mutually exclusive whose union is the sample space, and let B be an event compatible with $A_i$. Then the posterior probability, that $A_i$ occurs having occurred B will be: