Uniform Distribution
A probability experiment with a uniform probability distribution has a sample space (set of all possible outcomes)
for which each value is equally likely.
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That is, if there are 𝑛 possible values then 𝑃(𝑥) = 𝑛 for every 𝑥.
Example
1
When rolling a d6 die, the probability of each number in 𝑆 = {1,2,3,4,5,6} is exactly 6. If we consider the value of
each outcome to be it’s number, then the distribution is Uniform.
Here is the distribution represented graphically:
Probability Distribution for Rolling a d6 Die
1
Probability
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Value
Notice that each bar is the same height, so each value has the same probability.
A Non-Example
Suppose we roll two dice for our experiment instead, and add the numbers shown together to get a value. The set of
all possible values is {2,3,4, … ,12}, but not all values are equally likely:
Probability Distribution for Summing 2 d6 Dice
0.3
Probability
0.25
0.2
0.15
0.1
0.05
0
2
3
4
5
6
7
8
9
10
Value
This is NOT a uniform probability distribution (it doesn’t have a special name that I know of).
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12
Expected Value
Recall that 𝐸(𝑋) = ∑ 𝑥𝑃(𝑥) where 𝑥 is the value of an outcome.
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1
Because 𝑃(𝑥) = 𝑛 for every value 𝑥, we can factor out 𝑃(𝑥) = 𝑛 from the summation, giving
𝐸(𝑋) = 𝑃(𝑥) ∑ 𝑥
𝐸(𝑋) =
1
∑𝑥
𝑛
𝐸(𝑋) =
∑𝑥
𝑛
Notice that this is the average of the values in the sample space. In our first example (the d6 die), we have
6
1
𝐸(𝑋) = (∑ 𝑖 )
6
𝑖=1
=
1+2+3+4+5+6
6
= 3.5
Is this Uniform?
Which of the following experiments have a uniform distribution of outcomes?
A.
B.
C.
D.
E.
Choosing a card from a shuffled deck of cards and using the rank as the outcome
Choosing a student at random from our class and using the student’s age as the outcome
Choosing a customer at random in a restaurant and using their entrée order as the outcome
Rolling a die and using its parity (whether the value is even or odd) as the outcome
Rolling a die and using whether the value is 6 or not as the outcome.
I’ll tell you
A. Since the 13 values each occur 4 times in the deck, each outcome is equally likely. This is a uniform
distribution.
B. Since there are not an equal number of students at each age, some ages are more likely to occur than
others. This is not a uniform distribution.
C. Since some entrées are more popular than others, this is not a uniform distribution.
D. Since each die roll value is equally likely, and exactly half are even and half are odd, the TWO outcomes
(even and odd) are equally likely.
E. This outcome can be either true (yes, it’s a 6) or false (no, it’s not a 6). Although each roll value is equally
1
5
likely the two outcomes are true and false, which are not equally likely: 𝑃(𝑡𝑟𝑢𝑒) = 6, 𝑃(𝑓𝑎𝑙𝑠𝑒) = 6. So this
is not a uniform distribution.
Think about it
If a lottery had a uniform distribution of two outcomes (win and lose), what prizes could be possible for winning
tickets if the lottery was to be profitable? Would people play such a lottery?