Here's my spin.
You are playing horseshoes. When you throw your shoe you pretty much have three chances for an outcome. You completely miss the area and get 0 points. This is your worst case result. You get a ringer dead on is 3 points. This is your best case result. Or you can land somewhere in the middle. To make this easy without all the possible rules in horseshoes, we'll say you get 1 point.
Now in a triangular method you treat all points evenly. Meaning you really have no idea what result you could get when you throw the shoe. This is not very accurate, but you can say you will hit one of the three areas. Therefore all three points are treated equal. Which is why the calculation is to just add all three values and divide by three. And that's your best "guess" on what your score will be.
For the Beta distribution you want to take the same concept, but you have a bit of confidence in yourself and say "I will more times than not at least come close to the pin every time". Therefore your "most likely" spot to land is in the middle. So this is still a guess, but you have a stronger idea of where you'll land. Because of this, your middle number (in this case 1 point) you will hit more often. So they've said let's throw six times. and of those six, 4 of them should be in the middle. which is the same calculation as triangular but instead of 3 throws we take six and expect 4 of those in the middle.
When you're looking at this in a project, you are trying to predict risk and the outcome based on what you guess to be your worst, most likely and best shots. I'm selling a house. I can sell for 100 bucks at my minimum, 1000 bucks or ideally 10000 bucks. At that point you say "I have no clue" and do the triangular method (answer $3700) or I will most likely be in the 1000 range (because that's what my type of house usually sells for) and give that more weight as to what i can expect to make so you use the beta method ($2350).
If you know you will spend $2500 to buy it in the first place you might be walking into a mistake estimating on the triangular model and expecting to sell for $3700. Knowing you will be closer to the most likely gives you a closer estimate and you could be looking at losing money. Both are guesses, but one is putting more weight into that middle spot of "most likely". You could still sell it for $10,000 and be good regardless. These are just a tool used to put some values to your potential risk knowing very little other than best or worst case scenario.
This is how I've looked at the concept. I was hoping someone else would weigh in for you. But again. This is my take on it.
