Probability of object falling into a container

[stlag37]

On a manufacturing line moving uni-directionally at 60rpm (about 825 continers filled per minute) a bolt/nut has been observed in the bottom of a container.

The container neck (opening) is a 29mm diameter. The distance from the bottom of the machinery to the top of the bottle is 30cm. The bolt is 21mm long and is threaded through a nut 8mm in diameter. There are a total of 91 filling locations in one machine cycle.

What is the probability the bolt fell from the machine into the container? Can we hazard a guess as to if it was manually inserted into the container, versus accidentally falling into it?

Thanks!

 

[Janitor]

Are you allowed to assume that if the lower tip of the bolt breaks the plane of the imaginary disk at the opening of the bottle, the bolt will definitely drop into, and remain inside of, the bottle?

 

[stlag37]

Yes, if it fits in the bottle opening (breaking the diametrical plane) it will fall into the bottle and will not be able to, "bounce out."

 

[Janitor]

About all I can think is maybe you can treat it kind of like a cross-section problem, the way particle physicists do. Can you somehow calculate a ratio of bottle-opening area to total area, and if it is, say, one percent, then you can say there was just a 1% chance that the bolt would land in a bottle by chance? Maybe I am not even on the right track.

 

[HallsofIvy]

You need some sort of probability distribution for where the bolt would strike. Janitor was suggesting the simplest: the probability that the bolt will fall into the container is the area of the neck of the container divided by the area of the region on which such a bolt could fall. A more accurate distribution might be a 2-dimensional normal distribution about the container. Of course, you would have to include a probability that such a bolt would fall at all. Looks to me like there are simply too many unknowns to give a reasonable result.