Summary Basics of Probability

1. Basics of Probability
1.1 Probability Definitions

• Probability: A measure of the likelihood that an event will occur.
• Experiment: A process that leads to one of several possible outcomes.
• Sample Space (S): The set of all possible outcomes.
• Event (E): A subset of the sample space.

1.2 Calculating Probability

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
• Classical Probability: P(E)= 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠


• Empirical Probability: Based on observations or experiments.

• Theoretical Probability: Based on known possibilities.

1.3 Key Probability Rules

• Addition Rule: P(A∪B) =P(A)+P(B)−P(A∩B)



• Multiplication Rule: P(A∩B) =P(A)⋅P(B∣A) if A and B are dependent, P (A∩B)
=P(A)⋅P(B) if A and B are independent.



• Complementary Rule: P (𝐴𝑐 ) = 1-P(A)

1.4 Conditional Probability

• Conditional Probability:

P(A∩B)
P(A∣B) = 𝑃(𝐵)


1.5 Bayes' Theorem

• Bayes' Theorem:

P(B∣A)⋅P(A)
P(A∣B) = P(B)