Indicator Random Variables and the Fundamental Bridge

Indicators and the fundamental bridge

Counting with indicators

Patterns, runs, and a Ramsey surprise

Which identity is the fundamental bridge between probability and expectation?

10 balls drop independently into 4 equally likely boxes. What is the expected number of empty boxes?

In a room of 25 people with uniform independent birthdays over 365 days, the expected number of matching pairs is Binomial(25,2) times 1/365. Enter that value rounded to two decimals, like 0.85.

An event A has probability 0.2. What is the variance of its indicator? Enter a decimal like 0.16.

A random permutation of 1 through 12 has how many expected interior local maxima, at one third of the 10 interior positions? Enter like 3.33.

In your own words, state the most important definition, theorem or method of this lesson and explain what it says.