Independence and conditional probability

Learning Objectives Define conditional probability and independence in their own words. Calculate conditional probabilities using the formula P(A|B) = P(A and B) / P(B). Determine if two events are independent using the test P(A and B) = P(A) * P(B). Solve probability problems involving dependent events using the general multiplication rule. Interpret and calculate conditional probabilities from two-way frequency tables. Distinguish between independent and dependent events in real-world scenarios. If you know your friend aced their math test, does that change the probability that they studied for it? 🤔 Let's find out how new information changes the odds! This tutorial explores two fundamental concepts: conditional probability and independence. You will learn how the o...

TermDefinitionExample Conditional ProbabilityThe probability of an event (A) occurring, given that another event (B) has already occurred. It is denoted as P(A|B) and read as 'the probability of A given B'.The probability of a student passing a final exam given they attended all classes. The condition is 'attended all classes'. Independent EventsTwo events are independent if the outcome of one event does not influence the outcome of the other.Flipping a coin and getting 'heads', and rolling a standard six-sided die and getting a '4'. The coin flip does not affect the die roll. Dependent EventsTwo events are dependent if the outcome of the first event affects the outcome of the second event.Drawing a card from a deck and not replacing it, then drawin...

Formula for Conditional Probability P(A|B) = \frac{P(A \cap B)}{P(B)}, \text{ provided } P(B) > 0 Use this formula to find the probability of event A happening, given that you know event B has already happened. The denominator is always the probability of the 'given' event. General Multiplication Rule P(A \cap B) = P(A) \times P(B|A) \text{ or } P(A \cap B) = P(B) \times P(A|B) Use this to find the probability that two events both occur. It's especially useful for sequential events, like drawing two cards without replacement. Test for Independence \text{Events A and B are independent if and only if } P(A \cap B) = P(A) \times P(B) This is the definitive test for independence. If the probability of both events happening equals the product of their in...