Negative Binomial Distribution

Given random variable :

where is the number of trials required to observe the th success in a sequence of independent Bernoulli trials with a probability of success .

This is actually a generalization of the Geometric Distribution, where we wait for the th success rather than the first, but it's called the "negative Binomial Distribution" because we're counting the number of failures rather than successes.

An alternate formulation in terms of the geometric distribution:

If are independent and , then .

compare this with the binomial distribution PMF: .

(see choose)

Imagine we have (it takes 7 trials to reach 3 successes). Note that this is not the same as 3 successes in 7 trials. We require that the last trial be a success. So instead, we want 2 successes in 6 trials, and then a success in the last trial:

so we actually want the PMF of the binomial distribution for 2 successes in 6 trials AND (multiply) a success at the end, which brings us to:

Given :

A mathematician carries two matchboxes in his pockets. One in a pocket on the left and another on the right. Both matchboxes contain 40 matches. Due to the effects of smoking, this mathematician flips a coin to determine which matchbox he will get a match from whenever he wants to light a cigarette.

The mathematician opens the right matchbox to discover it empty, and the box on the left has 6 matches. What is the probability that this occurs?

solution
Suppose heads means right pocket and tails means left pocket.

• Since we used all 40 matches in the right pocket and then checked again, we have .
• Since we used all 40 matches in the right pocket and have 6 left in the left pocket, we have

In a large city, 10% of people have an O-positive blood type. If 100 people are picked at random:

1. Find P(10 have an O+ blood type)
2. Find P(10th O+ person is tested on the 100th sample)

solution
1.

We have . If is the number of trials until the 10th O+ person, then:

so we have:

We can see a pretty neat relation here. The second problem is asking for the first problem but only in a specific case that will occur 1/100 times, hence the probability is less by a factor of 10.

The Lakers are playing the Celtics in a 7-game series. Suppose the Lakers have a 60% chance of winning each game.

1. Find P(Lakers win exactly 6 games)
2. Find P(Celtics win the series)

solution
1.