Understanding Basic Probability

Study Notes

Probability indicates how likely an event is to occur
It is represented numerically between 0 and 1, where 0 signifies impossibility and 1 signifies certainty
Probability is widely applied across statistics, mathematics, science, and philosophy to infer likelihoods of potential occurrences and the mechanics of complex experiments

The probability of an event A is shown as P(A)
P(A) = n(A) / n(S), defines the probability of an event A where n(A) is the number of favorable outcomes for event A, and n(S) is the total number of possible outcomes in the sample space S
If a fair coin is tossed, the sample space S = {Head, Tail}, with the probability of getting a Head, P(Head) = 1/2, as there is one favorable outcome (Head) out of two possible outcomes.

Experiment: A process that yields well-defined outcomes
Sample Space: All possible outcomes of an experiment
Event: A subset of the sample space, representing a specific outcome or a group of outcomes
Outcome: A possible result of an experiment

Simple Event: An event consisting of only one outcome
Compound Event: An event consisting of more than one outcome
Independent Events: Events whose outcomes do not affect each other
Dependent Events: Events where the outcome of one affects the outcome of the other
Mutually Exclusive Events: Events that cannot occur at the same time

Any event's probability ranges from 0 to 1 (inclusive)
The probabilities of all possible outcomes in a sample space sum up to 1
For mutually exclusive events A and B, P(A or B) = P(A) + P(B)
For independent events A and B, P(A and B) = P(A) * P(B)
The complement of an event A (denoted A') includes all outcomes in the sample space not in A, and P(A') = 1 - P(A)

The likelihood of an event occurring given the occurrence of another event defines conditional probability
It is written as P(A|B), or "the probability of A given B"
Conditional probability is mathematically expressed as: P(A|B) = P(A and B) / P(B), given that P(B) > 0
P(A|B) indicates the probability of event A happening, knowing event B has already occurred
The occurrence of event B offers details that can change the likelihood of event A

Two events A and B are independent if one's occurrence doesn't impact the other's probability
If A and B are independent, then P(A|B) = P(A) and P(B|A) = P(B)
The probability of A occurring remains constant whether or not B has occurred, and vice versa

The theorem is a formula updating a hypothesis's probability based on evidence
It is a key concept in probability theory, used in statistics, machine learning, and decision theory
Bayes' Theorem is: P(A|B) = [P(B|A) * P(A)] / P(B)
P(A|B) represents the posterior probability of A given B
P(B|A) is the likelihood of B given A
P(A) is the prior probability of A
P(B) is the prior probability of B
The theorem is helpful for updating beliefs or probabilities with new evidence

If B1, B2, ..., Bn are mutually exclusive events and their union equals the sample space S, then for any event A
P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn)
This law calculates the probability of event A by considering all possible occurrences through different mutually exclusive events
It computes the overall probability by breaking it down into conditional probabilities based on different scenarios

Risk Assessment: Evaluating the probability of a specific event given certain risk factors
Medical Diagnosis: Determining the probability of a disease given certain symptoms
Machine Learning: Updating probabilities in Bayesian networks and classification algorithms
Finance: Assessing the probability of investment success or failure based on market conditions
Weather Forecasting: Predicting the likelihood of rain given current atmospheric conditions