Identify dependent and independent events

Learning Objectives Define independent and dependent events using precise mathematical language. Differentiate between scenarios involving sampling 'with replacement' and 'without replacement'. Analyze a real-world scenario to determine if two or more events are dependent or independent. Apply the formal multiplication rule, P(A and B) = P(A) * P(B), to test for independence. Apply the conditional probability test, P(B|A) = P(B), to test for independence. Explain why mutually exclusive events are always dependent. You're playing a board game and need to roll a 6 to win. Does your previous roll of a 3 affect your chances of rolling a 6 now? 🎲 Let's investigate! This tutorial will teach you how to distinguish between two fundamental types of eve...

TermDefinitionExample EventA specific outcome or a set of outcomes from a random experiment.In the experiment of rolling a six-sided die, an event could be 'rolling an even number', which corresponds to the set of outcomes {2, 4, 6}. Independent EventsTwo events are independent if the occurrence of one event does not affect the probability of the other event occurring.Flipping a coin and getting 'heads' (Event A) and rolling a die and getting a '5' (Event B). The coin flip has no impact on the die roll. Dependent EventsTwo events are dependent if the occurrence of one event changes the probability of the other event occurring.Drawing a King from a deck of cards (Event A), not replacing it, and then drawing another King (Event B). The first draw changes the co...

Test for Independence (Multiplication Rule) P(A \cap B) = P(A) \times P(B) Two events A and B are independent if and only if the probability that they both occur is equal to the product of their individual probabilities. This is the most common way to mathematically prove independence. Test for Independence (Conditional Probability) P(B|A) = P(B) \text{ or } P(A|B) = P(A) Two events A and B are independent if and only if the probability of B occurring, given that A has already occurred, is the same as the original probability of B. This shows that A's occurrence had no effect on B. Probability of Dependent Events P(A \cap B) = P(A) \times P(B|A) If two events are dependent, the probability of both occurring is the probability of the first event multiplied by the...