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# Category Archives: Geometric Distribution

## Rock Paper Scissors

A game of rock paper scissors consists of several rounds (players continue to play rounds until one player wins). In one round of rock paper scissors, two players each choose one of three options (rock, paper, or scissors). If they choose two different options, the game ends (rock beats scissors, scissors beat paper, and paper beats rock). If both players choose the same option, the game continues for another round. Assume each player chooses rock, paper or scissors uniformly at random and independently.

- What is the distribution of number of rounds played in a single game?
- What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
- Let \(n\) be the number of games played. How big must \(n\) be to ensure at least 100 rounds are played with 90% probability? Use an appropriate approximation to estimate.

## Almost geometric

An experimenter rolls a fair 6-sided die until they’ve seen both a 1 and a 2 (not necessarily consecutively). What is the experimenter’s expected number of rolls?

## Flipping Coins and Independence

An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.

The experimenter randomly selects one of the three coins and flips it until they get heads.

Let \(A\) be the event that the experimenter flipped the biased coin.

Let \(B\) be the event that it took the experimenter an even number of flips to get heads.

Are events \(A\) and \(B\) independent?

## Coin flipping game

Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let \(A\) be the event that you win. Let \(B\) be the event that you flip the coin 3 or more times. Let \(C\) be the event that you flip the coin 4 or more times.

- Compute \(\mathbb{P}(A)\).
- Are \(A\) and \(B\) independent?
- Are \(A\) and \(C\) independent?

## Memorylessness and the Geometric Distribution

Let \(X\) be a random variable with range \(\{0,1,2,3,\dots\}\) and distributed geometrical with probability \(p\).

- Show that for every \(n \geq 0\) and \(j\geq 0\),

\[ \mathbf{P}(X-n=j \mid X\geq n) = \mathbf{P}(X=j)\] - If \(X \) is the time to the failure of a machine, then \(\mathbf{P}( X\geq n) \) is the event that the machine has not failed by time \(n\). Why is the above property called
*Memorylessness*? - Show that the geometric distribution is the only random variable with range equal to \(\{0,1,2,3,\dots\}\) with this property.

## Expectation of geometric distribution

Compute the expectation of the geometric distribution using the fact that in this case

\(\mathbf{E}(X)= \sum_{k=1}^{\infty} \mathbf{Pr}(X\geq k) \)

## Introduction to Geometric random variables

Consider flipping a coin that is either heads (H) or tails (T), each with probability 1/2. The coin is flipped over and over (independently) until a head comes up. The outcome space is

\[ \Omega = \{H,TH,TTH,TTTH,\ldots\}. \]

(a) What is \( \mathbf{P}(TTH)\)?

(b) What is the chance that the coin is flipped exactly \(i\) times?

(c) What is the chance that the coin is flipped more than twice?

(d) Repeat the previous three questions for a unfair coin which has probability \(p\) of getting Tails.

[Author Mark Huber. Licensed under Creative Commons]