A probability experiment may be taken into consideration "fair" if every outcomes space equally likely, or (in part cases) if the expected value of part arbitrarily variable is 0 .
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example 1:
There room 6 football player in volleyball game. The team has to choose one of them randomly to it is in captain because that a game.
Tasha"s arrangement : Assign each player a number. Then roll a number cube. The captain is the player whose number come up.
Martin"s arrangement : Assign each player a number. Climate flip 3 coins. Pick a player according to the following chart.
HHH − 1 HHT − 2 HTH − 3 HTT − 4 THH − 5 THT − 6 TTH − 1 TTT − 2
inspect whether both the plans can be taken into consideration fair in picking a captain.
very first check Tasha"s plan for fairness.
The sample an are of the number cube is 1 , 2 , 3 , 4 , 5 , 6 and also each is same likely possible outcome.
every player has actually equal opportunity of choice as captain v probability that 1 6 .
Next examine Martin"s plan for fairness.
The sample an are of flipping 3 coins is HHH,HHT,HTH,HTT,THH,THT,TTH,TTT and there room 8 outcomes, which room equally likely.
The players 1 and also 2 have actually probability that 2 8 to be selected as captain, whereas the various other players have actually probability of 1 8 . Here, each does not have equal chance of selection as captain.
So, Martin"s arrangement cannot be considered to it is in "fair" in the selection of captain.
instance 2:
at a college fair, girlfriend are provided a variety of tokens. In one stall in ~ the fair, there is a spinner through 8 sectors. If the spinner lands on a red sector, you success 3 tokens. If friend land top top a environment-friendly sector, you win 5 tokens. If girlfriend land on any other sector, you lose 2 tokens.
Is this game fair?
The spinner has actually 8 sectors and each is equally most likely possibility.
Sample space is red sector, environment-friendly sector, 6 various other sectors
write the probability circulation for a single spin of spinner and also the quantity of tokens you win.
x = color Red eco-friendly others ( Blue and Yellow ) p ( x ) 1 8 1 8 6 8 Tokens 3 5 − 2
usage the weighted median 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 no zero, and the video game is no fair.
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Therefore you will lose about 0.5 tokens for a single spin.