# Using Probabilities to Make Fair Decisions

A probability experiment may be considered "fair" if all outcomes are equally likely, or (in some cases) if the expected value of some random variable is 0.

Example:

There are 6 players in volleyball game. The team has to choose one of them randomly to be captain for a game.

Tasha's plan : Assign each player a number. Then roll a number cube. The captain is the player whose number comes up.

Martin's plan : Assign each player a number. Then flip 3 coins. Select a player according to the following chart.

HHH - 1

HHT - 2

HTH - 3

HTT - 4

THH - 5

THT - 6

TTH - 1

TTT - 2

Check whether both the plans can be considered fair in selecting a captain.

First check Tasha's plan for fairness.

The sample space of the number cube is {1, 2, 3, 4, 5, 6} and each is equally likely possible outcome.

Each player has equal chance of selection as captain with probability of 1/6.

Next check Martin's plan for fairness.

The sample space of flipping 3 coins is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} and there are 8 outcomes, which are equally likely.

The players 1 and 2 have probability of 2/8 to be selected as captain, whereas the other players have probability of 1/8. Here, each does not have equal chance of selection as captain.

So, Martin's plan cannot be considered to be “fair” in the selection of captain.

Example:

At a school fair, you are given a number of tokens. In one stall at the fair, there is a spinner with 8 sectors. If the spinner lands on a red sector, you win 3 tokens. If you land on a green sector, you win 5 tokens. If you land on any other sector, you lose 2 tokens.

Is this game fair?

The spinner has 8 sectors and each is equally likely possibility.

Sample space is {red sector, green sector, 6 other sectors}

Write the probability distribution for a single spin of spinner and the amount of tokens you win.

Use the weighted average formula.

The expected value is not zero, and the game is not fair. So you will lose about 0.5 tokens for a single spin.