Calculate probabilities of events

Learning Objectives Define probability in the context of an infinite sample space. Apply the concept of limits at infinity to find the probability of an event over an infinite number of trials. Calculate the total probability of an event by summing an infinite geometric series. Evaluate the long-term or limiting probability of a system whose probability P(n) is a function of the number of trials, n. Solve word problems involving repeating trials by modeling them with limits and infinite series. Distinguish between the probability of an event in a finite number of trials and the limiting probability as trials approach infinity. What's the chance you'll *never* roll a 6 on a die, no matter how many times you try? 🎲 Let's use the power of calculus to find the su...

TermDefinitionExample Infinite Sample SpaceA set of all possible outcomes of an experiment where the number of outcomes is countably infinite.The set of positive integers {1, 2, 3, ...} representing the number of coin flips required to get the first 'Heads'. There is no upper limit to how many flips it could take. Limiting ProbabilityThe value that the probability of an event approaches as the number of trials or steps in a process approaches infinity.If the probability of a machine being operational after n days is P(n) = n/(n+1), the limiting probability is lim(n→∞) n/(n+1) = 1. This means the machine is almost certain to be operational in the long run. Geometric Distribution ProbabilityThe probability of the first success occurring on the k-th trial in a sequence of independe...

Sum of an Infinite Geometric Series S = \frac{a}{1 - r}, \quad \text{for } |r| < 1 Used to calculate the total probability when an event can occur on trial 1, OR trial 2, OR trial 3, etc., and the probabilities form a geometric sequence with first term 'a' and common ratio 'r'. The sum only converges to a valid probability if the absolute value of the ratio is less than 1. Limit of a Rational Function for Probability \lim_{n \to \infty} \frac{a_p n^p + ...}{b_q n^q + ...} = \begin{cases} \frac{a_p}{b_q} & \text{if } p=q \\ 0 & \text{if } p<q \\ \infty & \text{if } p>q \end{cases} Used to find the long-term probability when P(n) is a rational function of n (a polynomial divided by a polynomial). The limit is determined by the ratio of...