Mathematics Foundations/16.3 Conditional Probability

16.3 Conditional Probability 16.3.1 Definition of Conditional Probability Conditional probability measures the likelihood of an event occurring given that another event has already occurred. For two events and in a sample space , with , the conditional probability of given is defined as: where is the probability of both events and occurring, and is the probability of event occurring. 16.3.1.1 Interpretation The conditional probability can be interpreted as the probability of event when we restrict our sample space to only the outcomes in event . It represents an updated probability assessment when new information (the occurrence of event ) becomes available. 16.3.1.2 Properties of Conditional Probability 1. for any events and with 2. for any event with 3. for any event with 4. If are mutually exclusive events, then 16.3.2 Multiplication Rule The definition of conditional probability can be rearranged to give the multiplication rule: This can be extended to multiple events: 16.3.3 Law of Total Probability If forms a partition of the sample space (i.e., the 's are mutually exclusive and their union is ), and for all , then for any event : This can be extended to countably infinite partitions: 16.3.3.1 Graphical Representation The law of total probability can be visualized using tree diagrams, where branches represent different events and their conditional probabilities. 16.3.4 Bayes' Theorem Bayes' theorem provides a way to "invert" conditional probabilities. For events and with and : Using the law of total probability, if forms a partition of : 16.3.4.1 Terminology in Bayes' Theorem In the context of Bayes' theorem: is called the prior probability of is called the posterior probability of given is called the likelihood of given 16.3.4.2 Applications of Bayes' Theorem Bayes' theorem is widely used in: Medical diagnosis Machine learning and pattern recognition Spam filtering Legal reasoning Risk assessment 16.3.5 Independence of Events Two events and are independent if the occurrence of one does not affect the probability of the other. Mathematically, and are independent if: or equivalently This is equivalent to: 16.3.5.1 Conditional Independence Events and are conditionally independent given event if: Note that conditional independence given does not imply unconditional independence, and vice versa. 16.3.6 Monty Hall Problem The Monty Hall problem is a famous probability puzzle that illustrates conditional probability: A contestant on a game show is presented with three doors. Behind one door is a car, and behind the other two are goats. The contestant selects a door, and then the host, who knows what is behind each door, opens one of the other doors to reveal a goat. The host then asks the contestant if they want to switch their selection to the remaining unopened door. Should the contestant switch? Using conditional probability and Bayes' theorem, we can show that switching doors increases the probability of winning the car from to . 16.3.7 Exercises 1. A fair die is rolled. Let be the event that the number is even, and be the event that the number is greater than 3. Find and . 2. In a certain population, 1% of people have a specific disease. A test for this disease has a 95% chance of giving a positive result when a person has the disease and a 10% chance of giving a positive result when a person does not have the disease. If a person tests positive, what is the probability that they have the disease? 3. Two cards are drawn without replacement from a standard deck of 52 cards. What is the probability that the second card is an ace, given that the first card is an ace? 4. A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn in succession without replacement. What is the probability that both marbles are red? 5. Three coins are tossed. Given that at least two coins show heads, what is the probability that all three coins show heads?