Probability Fundamentals
Probability is a way of measuring how likely something is to happen. It is a measure of the likelihood of an event occurring.
Imagine you have a bag with colored balls in it. The probability of picking a red ball from the bag tells you how likely it is to get a red ball when you randomly reach in and grab one.
Probability is always given as a number between 0 and 1. If something has a probability of 0, it means it’s impossible. If it has a probability of 1, it means it’s certain to happen.
P(A)=Number of favorable outcomes/Total number of outcomes
Sample Space and Events:
Definition: The sample space represents the set of all possible outcomes of a random experiment, while events are subsets of the sample space. Example: In rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}, and events could be rolling an even number or rolling a number greater than 3.
P(rolling value 2)= 1/6
P(rolling an even number)= 3/6
P(rolling an odd number)= 3/6
Basic Probability Axioms:
Axioms of Probability: Probability measures must satisfy three basic axioms: non-negativity, normalization, and additivity.
- Non-negativity: Probabilities are non-negative values.
- Normalization: The probability of the entire sample space is 1.
- Additivity: The probability of the union of mutually exclusive events is the sum of their individual probabilities.
Probability is of two types:
- Objective-Based on statistics, experiments, and mathematical measurements
i. Classical probability- Based on formal reasoning about events with equally likely outcomes. Classical probability= (Number of desired outcomes/Total number of possible outcomes)
ii. Empirical probability- Based on previous results of experiment or past events. Empirical probability= (number of times a specific event occurs / total number of events)
Scenario: Drawing a card from a standard deck of 52 playing cards.
i. Classical Probability:
- Sample Space: There are 52 cards in a standard deck, and each card is equally likely to be drawn.
- Event: Drawing a heart (one of the four suits).
- Probability: P(heart)=Number of hearts/Total number of cards=13/52=1/4
- In classical probability, we assume that each card has the same chance of being drawn, resulting in a probability of 1/4 for drawing a heart.
ii. Empirical Probability:
- Observations: Draw cards from the deck multiple times and record the results.
- Event: Drawing a heart.
- Probability: P(heart)=Number of times a heart was drawn/Total number of draws
- For example, if you draw cards 100 times and get hearts 25 times, then the empirical probability of drawing a heart is 25/100=0.25.
- In empirical probability, the probability is based on the observed frequencies of drawing hearts from the actual draws.
In this scenario, classical probability assumes equally likely outcomes based on the structure of the deck, while empirical probability is based on observed frequencies from actual draws. Both approaches provide estimates of the likelihood of drawing a heart from the deck, but they differ in their underlying assumptions and methods of calculation.
Subjective Probability- Based on personal feelings, experience or judgement.
Conditional Probability:
Conditional probability quantifies the likelihood of an event occurring given that another event has already occurred.
Interpretation: It represents the updated probability of an event based on new information provided by the occurrence of another event.
Example: The probability of rain tomorrow given that the sky is cloudy represents conditional probability, as it considers the relationship between weather conditions.
Types of events:
Mutually exclusive events: Two events are mutually exclusive if they can not occur at the same time.
Independent events: Two events are independent if the occurrence of one event does not change the probability of other event.
Basic Rules of Probability:
Complement Rule: The complement rule in probability theory states that the probability of the complement of an event happening is equal to one minus the probability of the event happening. Mathematically, for an event A, the complement of A is denoted as A′ , and the complement rule can be expressed as: P(A′)=1−P(A).
Addition Rule: Applied to mutually exclusive events. The addition rule states that the probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection. P(A∪B)=P(A)+P(B)−P(A∩B)
Multiplication Rule: The multiplication rule in probability theory states that the probability of the intersection of two events is equal to the probability of the first event times the conditional probability of the second event given the first event has occurred. Mathematically, for two events A and B: P(A∩B)=P(A)×P(B∣A)
All other topics related to Probability will be coming soon.
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