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

Probability Level 2

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?

They gave the following answers:

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?

Andy Brian Charles None of them

2 solutions

Ajinkya Shivashankar
Dec 6, 2016

Method 1:-

$a_1>a_2$ probability is $\frac{1}{2}$

$a_1>a_3$ probability is $\frac{1}{2}$

As these 2 events are independent, answer is $\frac{1}{4}$

However, these 2 events are not independent.

Method 2:-

There are $3!=6$ possible orders out of which $1 \times 2!=2$ orders have 1st element as $a_1$

So probability is $\frac{2}{6}=\frac{1}{3}$

Method 3 (The shortest):-

There are 3 possible numbers which are equally likely to be the greatest. So the answer is 1/3.

I think the question could be improved by providing arguments for why the answer is 1/4 and 1/3, and then asking people to make their choice.

Calvin Lin Staff - 4 years, 6 months ago

I like that presentation! And in the solutions we can focus on explaining why one approach is flawed.

Christopher Boo - 4 years, 6 months ago

Grant Bulaong
Dec 8, 2016

Since they are all from the same interval, any of the three numbers could be the greatest. So just choose one.

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

2 solutions

0 pending reports