Probability problem

[Darya]

We randomly choose 2 numbers a,b from an interval a,b Є (0;2). What's the probability that a triangle with sides a,b,1 exists?

I don't understand how we can calculate probability since in that interval we have infinite numbers. Any ideas?

 

[lev888]

Easier problem: we randomly choose x from (0; 3). What's the probability that a triangle with sides 1, 1, x exists?

 

[pka]

We randomly choose 2 numbers a,b from an interval a,b Є (0;2). What's the probability that a triangle with sides a,b,1 exists? I don't understand how we can calculate probability since in that interval we have infinite numbers. Any ideas?

Think: Triangle inequality.
Look Here What does that have to do with this problem?

 

[Dr.Peterson]

if you've only learned about discrete probability, this would be difficult. One way to think about this is graphical: make a square in which x is between 0 and 2, and y is between 0 and 2. What points in this square represent pairs of numbers for which the triangle can be made? The probability is the ratio of the area of that region to the area of the whole square, because any point in the square is equally likely to be chosen.

 

[Darya]

Easier problem: we randomly choose x from (0; 3). What's the probability that a triangle with sides 1, 1, x exists?

Thank you for answering and sorry for the late reply. X must lie in the interval (0;2), right? So is it 2/3?

 

[Dr.Peterson]

Please show the square you drew, or whatever other work you did. You have not shown any reason for your answer of 2/3.

As I said, this is very difficult if you have never seen anything like it. It will help if you tell us what you have learned, and why you are working on this problem, which surely would not have been assigned to a class you are taking. We need to know what we can assume you know, in our answers.

 

[lev888]

Thank you for answering and sorry for the late reply. X must lie in the interval (0;2), right? So is it 2/3?

Correct, but please see post #6.

 

[Darya]

Please show the square you drew, or whatever other work you did. You have not shown any reason for your answer of 2/3.

As I said, this is very difficult if you have never seen anything like it. It will help if you tell us what you have learned, and why you are working on this problem, which surely would not have been assigned to a class you are taking. We need to know what we can assume you know, in our answers.

I'm only in high school and we really were never given such problems. Those are from entering exams of a university abroad and I see what we learn is far not as advanced. This is though the only problem I have difficulty with for now.

I'm not sure where that dividing line should be.

 

[Dr.Peterson]

I'm only in high school and we really were never given such problems. Those are from entering exams of a university abroad and I see what we learn is far not as advanced. This is though the only problem I have difficulty with for now.

I'm not sure where that dividing line should be.

Thanks for confirming my guesses about your background.

You have the right idea, but the triangle inequality works for each of the sides of the triangle. If the sides are a, b, and 1, then

a + b > 1
a + 1 > b
b + 1 > a

Graph all three of these (the one you drew is not quite right), and you'll be on the right track.

 

[Darya]

Thanks for confirming my guesses about your background.

You have the right idea, but the triangle inequality works for each of the sides of the triangle. If the sides are a, b, and 1, then

a + b > 1
a + 1 > b
b + 1 > a

Graph all three of these (the one you drew is not quite right), and you'll be on the right track.

Got it! The area of the square is 4, the area of the is 4-3*0,5=2,5. So it's 5/8. Thanks a lot for your patience and help!

 

[Dr.Peterson]

Sounds like you probably have the right picture.

Now, if you'd like to learn more about this kind of problem, search for Geometric Probability (but not the geometric probability distribution).

Here is one good source, with many examples: https://brilliant.org/wiki/1-dimensional-geometric-probability/

Here is a shorter introduction: https://www.cut-the-knot.org/Probability/GeometricProbabilities.shtml

 

[Darya]

Sounds like you probably have the right picture.

Now, if you'd like to learn more about this kind of problem, search for Geometric Probability (but not the geometric probability distribution).

Here is one good source, with many examples: https://brilliant.org/wiki/1-dimensional-geometric-probability/

Here is a shorter introduction: https://www.cut-the-knot.org/Probability/GeometricProbabilities.shtml

I couldn't be thankful enough!