Geometric and Negative Binomial, the Waiting Distributions

The Geometric story

Expectation by first-step analysis

Negative Binomial and the r-th success

Memorylessness

A shooter hits with probability 0.2 per independent shot. What is the probability the first hit takes more than 5 shots?

A website gets an order with probability 0.1 each independent minute. Forty quiet minutes have passed. What is the probability the next order takes more than 10 more minutes?

A rare event has probability 0.004 per independent trial. On average, how many trials until the first occurrence? Enter a number like 250.

Waiting for the 3rd success with per-trial probability 0.25, the expected total number of trials is r/p. Enter that integer.

Collecting coupons, you have 4 of 5 kinds. Each box independently gives any kind with probability 1/5. The expected number of boxes to get the last missing kind is what number? Enter like 5.

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