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**Unformatted text preview: **tisfy
∞
∞
∞
P
[ai , bi ] =
P {[ai , bi ]} =
( b i − ai ) .
i=1 i=1 i=1 For instance, the probability that the outcome is in the interval [0, 1/4] is 1/4, the probability
the outcome is in the interval [1/3, 1/2] is 1/6, and the probability that the outcome is in
either the interval [0, 1/4] or [1/3, 1/2] should be 1/4 + 1/6 = 5/12. That is,
P {[0, 1/4] ∪ [1/3, 1/2]} = P {[0, 1/4]} + P {[1/3, 1/2]} = 11
5
+=.
46
12 If P is to be the uniform probability on [0, 1], then it should also be unaﬀected by shifting.
In particular, it should only depend on the length of the interval and not the endpoints
themselves. For instance,
1
P {[0, 1/4]} = P {[1/6, 5/12]} = P {[3/4, 1]} = ,
4
or, more generally, 1
for every 0 < r ≤ 3/4.
4
Note that if 3/4 < r < 1, then [r, 1/4 + r] is no longer a subset of [0, 1]. But if we allow
“wrapping around” then [r, 1/4 + r] might become two disjoint intervals, each a subset of
[0, 1], having total length 1/4. For instance, if...

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