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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 1:

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 2:

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?

Math diagram

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.

x = color Red Green Others ( Blue and Yellow ) P ( x ) 1 8 1 8 6 8 Tokens 3 5 2

Use the weighted average formula.

E ( x ) = 3 1 8 + 5 1 8 + ( 2 ) 6 8 = 3 8 + 5 8 12 8 = 4 8 or 0.5

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

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