Probability of Transcendental

Probability Level 3

If we pick a number uniformly at random from $[0,1]$ , what is the probability that the number is a transcendental number ?

Inspiration, see solution .

0

\frac{1}{3}

1

\frac{1}{2}

1 solution

Brian Charlesworth
Jan 22, 2016

Algebraic numbers and transcendental numbers form a partition of the interval $[0,1]$ . As the algebraic numbers are countable, they have a measure of $0$ , and since the respective measures of the algebraic and transcendental numbers are complementary and add to $1$ the measure of transcendental numbers on the given interval is $1$ , i.e., the probability that a number picked at random on $[0,1]$ is transcendental is $\boxed{1}$ .

Doesn't "probability of picking a transcendental in $[0,1]$ is $1$ " mean that picking of a transcendental, in this case, is a sure event (which I suppose, is not !) ? I am perplexed.

Venkata Karthik Bandaru - 5 years, 4 months ago

"Probability of 1" is different from "surely happens".

Ivan Koswara - 5 years, 4 months ago

Courtesy: Wikipedia

Probability is the measure of the likelihood that an event will occur.[1] Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty)

I see that the above statement complies with "Probability of 1" meaning "surely happens".

Venkata Karthik Bandaru - 5 years, 4 months ago

@Venkata Karthik Bandaru – That notion is only correct for finite event space. See the concept of almost surely .

Ivan Koswara - 5 years, 4 months ago

@Ivan Koswara – Woah, now I get it. Thanks !

Venkata Karthik Bandaru - 5 years, 4 months ago

Probability of Transcendental

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