Lecture 26 – Probability

You may want to download the lecture slides that were used for these videos (PDF).

1. Foreword & Some Motivating Examples

Here, we give examples of real-world applications of probability.  Many of our examples will deal with games of chance and the notion of “gambling”. This approach is historically accurate as a good fraction of early interest in probability has its roots in this setting. But, this discussion is not intended in any way to be an endorsement of gambling. (11:24)

2. Finite Probability Spaces (1)

In this video we define a finite probability space, Ω. (1:50)

3. Finite Probability Spaces (2)

If X = {x1, x2, …, xn}, then it is natural to abbreviate P({xi}) as P(xi). So to define the probability measure P, it is enough to know the values of P(xi) for each i = 1, 2, …, n. (3:03)

4. Finite Probability Spaces (3)

Five cards are selected at random from a standard deck of 52 playing cards. (11:20)

5. Finite Probability Spaces (4)

An opaque jar contains 5 red marbles, 4 blue marbles and 7 green marbles. Two marbles are selected at random. What is the probability that both are red? (5:50)

6. Bernoulli Trials

Suppose that an experiment either “succeeds” or it “fails”. The probability of success is p and the probability of failure is q = 1 – p. The experiment is then repeated n times with different trials unaffected by the outcomes of the other trials. (10:34)

7. Infinite Probability Spaces – Examples

Example: In a Bernoulli trial set up, there are three outcomes x1, x2, and x3 with probabilities p1, p2, and p3, respectively. The trial is repeated until either x1 occurs (this is a win) or x2 occurs (this is a loss). As long as x3 occurs, the trial is repeated. What is the probability of a win? (6:10)

8. Random Variables & Expectation

In this video we define random variables and expected values. (6:47)

9. Returning to Our Motivating Examples

In this example, we return to our motivating examples that we gave at the start of the lecture and see if we can answer them. (5:23)

10. One Final Comment

Why do people gamble? Perhaps people gamble while knowing irrefutable mathematical facts just for the thrill that comes with occasional jackpot, or they support the ways that the state promises to spend the profits they make. (6:46)