Independence and conditional probability

Learning Objectives Define conditional probability and independence in the context of a variable parameter 'n'. Set up probability expressions as functions of a variable 'n' that represents a growing population or number of trials. Calculate the limit of a conditional probability P(A|B) as n approaches infinity. Use limits to determine if two events become independent as the number of trials or population size approaches infinity (asymptotic independence). Apply the rules for limits of rational functions to solve complex probability problems. Interpret the meaning of a limiting probability of 0, 1, or a value in between in the context of a real-world scenario. What's the real chance of picking a specific person out of a crowd if the crowd size grows...

TermDefinitionExample Conditional ProbabilityThe probability of an event (A) occurring, given that another event (B) has already occurred. It is denoted by P(A|B).In a deck of 52 cards, the probability of drawing a King, given that you've already drawn a Queen (and not replaced it), is P(King|Queen) = 4/51. Independence of EventsTwo events are independent if the occurrence of one does not affect the probability of the other occurring. Mathematically, P(A and B) = P(A) * P(B).Flipping a coin twice. The outcome of the first flip (Heads) does not affect the probability of the second flip being Heads. P(Heads on 1st and Heads on 2nd) = (1/2) * (1/2) = 1/4. Limit at InfinityThe value that a function f(n) approaches as the variable 'n' increases or decreases without bound. It des...

Conditional Probability Formula P(A|B) = \frac{P(A \cap B)}{P(B)} Use this formula to find the probability of event A happening, on the condition that event B has already happened. P(A ∩ B) is the probability that both A and B occur. Test for Independence P(A \cap B) = P(A) \cdot P(B) Two events A and B are independent if and only if the probability of them both occurring is equal to the product of their individual probabilities. An equivalent test is P(A|B) = P(A). Limit of a Rational Function at Infinity \lim_{n \to \infty} \frac{a_k n^k + \dots}{b_m n^m + \dots} = \begin{cases} 0 & k < m \\ \frac{a_k}{b_m} & k = m \\ \pm\infty & k > m \end{cases} To find the limit of a ratio of polynomials as n -> infinity, compare the degrees of the numerator...