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Toddzilla · 03-10-2021, 06:44 PM

So let's approach it like this......

Imagine the stick has a mark in the very middle, point m

The first mark on the stick is on one side of m. That's what we're calling piece A.

Half A is 1/2 the stick with a mark on it and Half B is 1/2 the stick with no mark.

A second mark is made.

If the second mark is on Half A, we cannot make a triangle (as determined a few ways above.)

The chances of a second mark being on the same Half as the first mark is....50%.

If the second mark is on Half B, then we MAY be able to have a triangle.

So the chances we can make a triangle are demonstrably LESS than 50%.

Imagine the stick now a line with 5 points: The "beginning" of the stick, the first mark, the middle, the second mark, and the "end" of the stick.

In order for us to be able to make a triangle, the length of the stick between the first mark and second mark must be less than 1/2 the length of the stick.

Another way to say that is this: The distance between the "beginning" of the stick and the first mark must be GREATER THAN the distance between the middle of the stick and the second mark. Otherwise the distance between the first and second marks is greater than 1/2 the length of the rod, which is a no-go.

Now it's a matter or probabilities and limits, which I'll try not to overdo.

On Half A, there is equal probability of the first mark being anywhere on Half A.

If the first mark is VERY CLOSE to the "beginning" of the stick, the odds the second mark is CLOSER to the halfway mark is VERY SMALL. As the first mark gets closer and closer to the "beginning" of the stick, those odds approach 0%

If the first mark is VERY FAR AWAY from the "beginning" of the stick, the odds the second mark is CLOSER to the halfway mark is VERY LARGE. As the first mark gets further and further away from the "beginning" of the stick (i.e. as it approaches the middle), those odds approach 100%.

The average of 0% and 100% is 50%

and

50% (the chances the second mark is closer to the middle than the first mark is to the "beginning" of the stick) of 50% (the chances the second mark is not on the same half as the first mark) is 25%

03-10-2021, 06:44 PM	#47
Toddzilla Pro Starter Join Date: Jan 2001 Location: Burke, VA	So let's approach it like this...... Imagine the stick has a mark in the very middle, point m The first mark on the stick is on one side of m. That's what we're calling piece A. Half A is 1/2 the stick with a mark on it and Half B is 1/2 the stick with no mark. A second mark is made. If the second mark is on Half A, we cannot make a triangle (as determined a few ways above.) The chances of a second mark being on the same Half as the first mark is....50%. If the second mark is on Half B, then we MAY be able to have a triangle. So the chances we can make a triangle are demonstrably LESS than 50%. Imagine the stick now a line with 5 points: The "beginning" of the stick, the first mark, the middle, the second mark, and the "end" of the stick. In order for us to be able to make a triangle, the length of the stick between the first mark and second mark must be less than 1/2 the length of the stick. Another way to say that is this: The distance between the "beginning" of the stick and the first mark must be GREATER THAN the distance between the middle of the stick and the second mark. Otherwise the distance between the first and second marks is greater than 1/2 the length of the rod, which is a no-go. Now it's a matter or probabilities and limits, which I'll try not to overdo. On Half A, there is equal probability of the first mark being anywhere on Half A. If the first mark is VERY CLOSE to the "beginning" of the stick, the odds the second mark is CLOSER to the halfway mark is VERY SMALL. As the first mark gets closer and closer to the "beginning" of the stick, those odds approach 0% If the first mark is VERY FAR AWAY from the "beginning" of the stick, the odds the second mark is CLOSER to the halfway mark is VERY LARGE. As the first mark gets further and further away from the "beginning" of the stick (i.e. as it approaches the middle), those odds approach 100%. The average of 0% and 100% is 50% and 50% (the chances the second mark is closer to the middle than the first mark is to the "beginning" of the stick) of 50% (the chances the second mark is not on the same half as the first mark) is 25% Last edited by Toddzilla : 03-10-2021 at 06:45 PM.