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Fair/ Bias
Expected Value
A biased die is used in a game. The probabilities of getting the
six different numbers on the die are shown in the table below.
Expected Value E(X) is the
average outcome of an event
Number
Probability
𝐸 𝑋 =
𝑥. 𝑃(𝑥)
To find, we multiply every
possible outcome by the
probability for that outcome
and then add all these values
together.
1
2
3
4
5
6
0.25
0.25
0.15
0.15
0.1
0.1
On a ‘fair’ die, every number has
an equal chance of being rolled
1
𝑜𝑟 0.166
6
This ‘bias’ die is more likely to
land on a 1 or 2 than a 5 or 6.
0.25 > 0.1
Find the expected value of the random variable X, where X is
the number thrown.
We make a table and multiply each outcome by the probability
of that outcome.
Playing Value Games
X
P(X)
X.P(X)
If we played a game with this die
1
0.25
0.25
where we won the € amount of the
number we rolled the game would be
2
0.25
0.5
worth playing if it cost less than €2.90.
3
0.15
0.45
4
0.15
0.6
5
0.1
0.5
6
0.1
0.6
Add all the
values in X.P(X)
𝑥. 𝑃(𝑥)
2.9
The expected value when rolling this ‘biased’ die is 2.9
It would be bad value if the game cost
more than €2.90.
For example if this game cost €3 to
play we would lose on average €0.10
ever time we played.