# Is it 1/3 or is it 1/4?

When learning about probability, Alex, Brian and Charles were asked the following question:

Choose 3 numbers uniformly at random: $$a_1, a_2, a_3 \sim [0,1]$$. What is the probability that $$a_1$$ is greatest?

Andy: $$a_1$$ is either the greatest or not the greatest. Hence the probability is $$\frac{ \text{ positive outcomes}} { \text{ total outcomes} } = \frac{1}{2}$$.

Brian: One of $$a_1, a_2, a_3$$ is the greatest. Hence the probability is $$\frac{ \text{ positive outcomes}} { \text{ total outcomes} } = \frac{1}{3}$$.

Charles: $$a_1$$ is either greater or less than $$a_2$$ with probability $$\frac{1}{2}$$. Similarly, $$a_1$$ is either greater or less than $$a_3$$ with probability $$\frac{1}{2}$$. Hence, $$a_1$$ is greater than $$a_2$$ and $$a_1$$ is greater than $$a_3$$ with probability $$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$.

Who is correct?

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