Introduction To Applied Probability

Students learn basic probability definitions, including random experiments, outcomes, and the sample space. The lecture uses examples of unbiased coins, dice, and 52-card decks to illustrate possible results.

Define elementary events and the sample space, and explain equally likely outcomes. Use a six-faced die and coin toss to illustrate independent events and even and odd results.

Explore basic probability definitions, including complementary events, mutually exclusive sets, intersections and unions, sample space, and exhaustive event sets, with die-based examples.

Explore mutually exclusive and exhaustive events, unions and intersections, and sample space concepts with examples of even and odd numbers; note independence and complement relationships.

Explore the classical definition of probability, including the sample space, events and complements, and apply addition and multiplication rules through three solved examples: coins, primes, and critics.

Apply the mathematical definition of probability to two fair dice, find the sample space of 36 outcomes, and show the event’s probability as 11/12 for a total less than 11.

Explore probability definitions and solved examples 5 and 6, focusing on sample space. Compute card-drawing probabilities, such as obtaining a king, a queen, and a number, using the deck's cards.

Explore the definition of probability and reliability through a solved example, showing how to count the sample space and favorable outcomes to compute event probability.

Learn essential probability symbols for events A, B, C, including union, intersection, and complements, and analyze outcomes like at least one, exactly one, and at least two of the three.

Explore core probability results, including unions, intersections, complements, and inclusion-exclusion, with an example using A, B, and C to illustrate mutually exclusive and exhaustive events and independence.

Using independence and complements, determine P(A∩B)=1/12 and P(A^c∩B^c)=1/2, then find P(A) and P(B) from P(A)+P(B)=7/12 and P(A)P(B)=1/12, yielding 1/3 and 1/4.

Analyze a solved probability example of a student’s success with events A, B, and C, deriving a relation between p and q and exploring complements, unions, intersections, and independence.

Compute the probability that at least one of A, B, or C occurs using exact-one and intersection–union identities, proving P(A ∪ B ∪ C) > 1/2.

Derive key probability results from solved examples 5 and 6 by applying union and intersection of events A, B, and C to obtain bounds.

Explore why the intersection of events may differ from the product of probabilities, clarifying when events are independent or not in a solved example.

Learn conditional probability, its formula P(A|B) = P(A∩B)/P(B), and how independence, union, and complement affect probabilities, illustrated by a two-die example yielding P(sum=9 | first die is 5) = 1/6.

Study conditional probability through a solved example, defining a sample space, events, and the probability of failing mathematics given prior outcomes.

Apply conditional probability by using the intersection of events and the probability formula to solve solved examples and generalize the approach for different event scenarios.

Apply the theorem of total probability to a production scenario with machines A, B, and C. Compute the defective probability using exhaustive, mutually exclusive events and conditional probabilities.

Explore the theorem of total probability through a solved example with urn transfers, white and black balls, and conditional calculations.

Apply the theorem of total probability to solved example 3 on sequential drawing with white and black balls, deriving the probability of drawing a black ball.

Apply the theorem of total probability in solved example 4 to compute the probability a company lasts one year using reliability events and exclusive, exhaustive conditions.

Apply Bayes' theorem to compute posterior probabilities using prior and conditional probabilities, illustrated with a solved bag-and-ball example and checks for mutually exclusive and exhaustive events.

Apply Bayes' theorem to a solved example with a fair die (prior 1/6) and a truthfulness of 3/4; find P(actual six | says six) as 3/8.

Explore Bayes' theorem through solved example 3, assessing guessing probability 1/3 and knowing probability 1/6 in a four-choice question. Determine the probability he knew the answer given a correct response.

Apply Bayes' theorem to a scenario with independent witnesses, examining how truth and lies affect probabilities and the complement of intersections under event independence.

Explore Bernoulli trials as independent experiments with fixed success probability, and apply to solved examples, including computing exact numbers of heads and the probability of even head counts.

Explore Bernoulli's trials through a solved example: compute the probability that at least four of five ships reach safely using a binomial framework.

Explore a solved Bernoulli's trials example where a man takes 11 forward or backward steps and ends one step from the start, using binomial coefficients and combinations.

explores bernoulli's trials through a solved example with coin tosses, examining independence, mutual exclusivity, and computing the probability of two players achieving the same number of heads.

Analyze conditional probability in a two-child family using Bernoulli's trials, comparing at least one boy to exactly one boy and deriving the probability that both children are boys.

Explore Bernoulli's trials generalisation through the concept of exhaustive and mutually exclusive events in random experiments, and work through a solved example to compute probabilities over multiple trials.

Present a geometric probability example in uncountable uniform spaces, where two independent random times x and y in a window meet if |x−y|<20, using a sample space and area reasoning.

Explore uncountable uniform spaces through a solved example involving random points in a plane, a square, and a circle, calculating corner-based distance regions and their probabilities.