Basic Probability Aspects and Bayesian Theorem Quiz

The forecast is 70% chance of rain today. So, take my ______

It’s cloudy, it may rain. So, take my ______

Objective Probability is based on ___________ data.

Subjective Probability is based on ___________ experiences.

Mutually exclusive events have ___________ outcomes on any one trial.

Collectively exhaustive events list all ___________ outcomes.

Two dice are rolled, find the probability that the sum is equal to 1: 0/36 or ___________

Two dice are rolled, find the probability that the sum is equal to 4: 3/36 or ___________

A die is rolled, and a coin is tossed, find the probability that the die shows an odd number, and the coin shows a ___________

Bayesian Theorem is used to calculate ___________ probability.

Probability Concepts

Mutually exclusive events: cannot occur simultaneously
Non-mutually exclusive events: can occur simultaneously
Statistically independent events: occurrence of one event does not affect the probability of another event
Statistically dependent events: occurrence of one event affects the probability of another event

Marginal and Joint Probability

Marginal probability: probability of an event occurring (e.g., P(A))
Joint probability: probability of two or more events occurring at the same time (e.g., P(A and B))
Conditional probability: probability of an event occurring given that another event has occurred (e.g., P(A|B))

Bayesian Theorem

Bayes' Theorem: used to update probabilities based on new information
Formula: P(A|B) = P(B|A) x P(A) / P(B)
Posterior probability: revised probability after considering new information
Prior probability: initial probability before considering new information

Example Problems

Probability of two girls given at least one girl: 1/3 or 0.33
Probability of a fair die given a roll of 3: 0.22
Probability of a loaded die given a roll of 3: 0.78

Fair and Loaded Dice

Fair assumption: uniform or equal prior probabilities assigned to all hypotheses
Loaded die: non-uniform probabilities assigned to outcomes
P(Fair) = 0.50, P(Loaded) = 0.50
P(3|Fair) = 0.166, P(3|Loaded) = 0.60

Test your knowledge on basic probability aspects such as addition of mutually exclusive and non-mutually exclusive events, statistically independent and dependent events, marginal/simple probability, joint probability, conditional probability, as well as Bayesian Theorem. Explore how Bayes' theorem is used to incorporate additional information in probability calculations.