Find conditional probabilities

Learning Objectives Define conditional probability in the context of continuous random variables over infinite intervals. Set up improper integrals to represent probabilities of events like P(X > a). Apply the formula P(A|B) = P(A ∩ B) / P(B) to events defined by inequalities. Construct a limit expression for a conditional probability as a variable parameter approaches infinity. Evaluate limits of conditional probabilities using algebraic simplification or L'Hôpital's Rule. Interpret the result of a limiting conditional probability in the context of a given scenario, such as component lifetime or waiting times. If a satellite has already operated for 10 years, what's the probability it will survive another year? What if it has already operated for 100 years...

TermDefinitionExample Conditional ProbabilityThe probability of an event (A) occurring, given that another event (B) has already occurred. It is denoted P(A|B).The probability that a card drawn is a King, given that it is a face card. Continuous Random VariableA variable that can take on any value within a given range. Its probability is described by a continuous function rather than a set of discrete points.The exact height of a student, the lifetime of a lightbulb, or the waiting time for a bus. Probability Density Function (PDF)A function, f(x), used to describe the probabilities for a continuous random variable. The probability of the variable falling within a particular range is the integral of this function over that range.For a variable X, P(a ≤ X ≤ b) = ∫[a,b] f(x) dx. The total a...

Formula for Conditional Probability P(A|B) = \frac{P(A \cap B)}{P(B)} This is the fundamental definition of conditional probability. To find the probability of A given B, we divide the probability of both A and B happening by the probability of B happening. Probability as an Improper Integral P(X > a) = \int_{a}^{\infty} f(x) \,dx For a continuous random variable X with PDF f(x), the probability that X is greater than some value 'a' is found by integrating the PDF from 'a' to infinity. Limiting Conditional Probability \lim_{t \to \infty} P(X > t+k | X > t) = \lim_{t \to \infty} \frac{\int_{t+k}^{\infty} f(x) \,dx}{\int_{t}^{\infty} f(x) \,dx} This structure combines the previous two rules to answer questions about long-term conditional be...