A 10-cm stick has a mark at each centimeter. By breaking the stick at two of these nine marks at random, the stick is split into three piece

Hi Chris, hot18594, Ginger and guest.    And thanks for your answers and interaction   

I think that we are all in agreement that there are 36 different possible combinations of cuts.  

9 places to cut, cut in 2 places   9C2 = 36

We are also in agreement that the sides of the triangle will be (2,4,4) or (3,3,4) units in length.

BUT where can the CUTS happen for this??

I will look at 2,4,4

1 cut 3,4,5, cut, 7,8,9,10     that would give 2,4,4

or

1,2,3,cut,5,cut,7,8,9,10      that would give 4,2,4

or

1,2,3,cut,5,6,7,cut,9,cut      that would give 4,4,2

So there are 3 ways to cut the stick to get a trinagle of sides  2,4,4

there will also be 3 ways to cut the stick to get  3,3 and 4

So there are 6 favourable ways to cut the stick.

Probability of being able to make a triangle will be    6/36   =    1/6

I think that is what Ginger was refering to.  

There are two ways to choose the lengths to be trianlges, (2,4,4) and (3,3,4).  The number of ways of choosing the break points is C(9,2) = 36.  So the probability is 2/36 = 1/18.  Hope that helps!

Got it, its 1/6