The Ball Bag problem

[Jon]

I have been trying to sort out this little problem for a while now (if fact ever since I did my PhD 20 years ago!) The solution is probably simple but I am not a mathematician and just curious about a solution.

I have been given a bag full of balls. The balls are of different sizes but are all spherical and I know that the smallest is 5cm in diameter.
I take a cross section of the bag through the center - and get a pattern of a large outline (the bag) and many smaller circles inside (the balls) some of which are touching.

My question is this:
What information can I get from analysing the pattern? Can I tell:
The number of balls in the bag?
The average diameter?
How many of each size I have? ie ball size distribution
The largest ball in the bag?

I'm guessing a lot of this is probability? ie what are the chances of cutting the largest ball exactly through its middle. If all the balls were one size would a section give a normal distribution of diameters?

Many thanks.

[GeradHum]

Analyzing the cross-sectional pattern of the bag with balls can provide some information, but it may not allow you to determine all the details you're interested in with certainty. Here's what you can infer and what you can't infer:

Number of Balls in the Bag: You cannot accurately determine the exact number of balls in the bag just from the cross-sectional pattern. If the balls are tightly packed, you might see a dense arrangement of circles in the cross-section, but you won't know how many layers of balls there are vertically.

Average Diameter: It's difficult to determine the precise average diameter from a single cross-section because you might cut through some balls off-center, making them appear smaller than their actual diameter, while others may be cut through the center. To estimate an average diameter, you'd need multiple cross-sections from different angles.

Ball Size Distribution: You can make some qualitative observations about the distribution of ball sizes. If you see a wide range of circle sizes in the cross-section, you can infer that there's a variety of ball sizes in the bag. However, you won't be able to quantify this distribution without additional information or more cross-sections.

Largest Ball: It's unlikely that you can determine the size of the largest ball in the bag from a single cross-section, especially if the cut doesn't pass through its center. The largest ball might not even be visible in that particular cross-section.

Probability and Normal Distribution: The cross-sectional pattern might appear somewhat like a normal distribution if all the balls were the same size and the cut was made randomly through the bag. However, in reality, with different-sized balls and non-random cutting, the cross-section will not provide a perfect normal distribution.

In summary, while you can make some qualitative observations about the ball sizes and their arrangement in the bag from a single cross-section, you cannot precisely determine the number of balls, the average diameter, the size distribution, or the size of the largest ball without more information or a different approach, such as measuring or counting the balls directly. Multiple cross-sections from different angles might provide a better estimation of these parameters, but it may still be challenging to obtain precise values.

[Guest]

Analyzing the cross-section of the bag can provide some information about the distribution of ball sizes, but certain details might be difficult to ascertain precisely without additional data or assumptions. Here's what can potentially be determined:

Number of Balls in the Bag: If you can accurately count the number of smaller circles (balls) in the cross-section, you would have an estimate of the number of balls in the bag. However, this might not be exact due to variations in how the balls are arranged and whether any are obscured from view.

Average Diameter: By measuring the diameters of the smaller circles (balls) in the cross-section and taking their average, you can estimate the average diameter of the balls in the bag. Again, this would be an estimate and might not precisely represent the true average diameter due to variations in arrangement and visibility.

Ball Size Distribution: Analyzing the diameters of the smaller circles (balls) in the cross-section can provide insights into the distribution of ball sizes. If you see a range of diameters, you can infer that there is a variety of ball sizes in the bag. However, determining the exact distribution would be challenging without additional data.

Largest Ball in the Bag: It's difficult to determine the largest ball in the bag solely from a cross-section. Even if you observe a large circle in the cross-section, it might not necessarily correspond to the largest ball because the largest ball may not have been cut perfectly through its middle. Determining the largest ball would likely require additional information or assumptions about the arrangement of balls within the bag.

As you mentioned, the analysis involves probability and assumptions about the arrangement of balls within the bag. If the balls were uniformly distributed and randomly arranged, you might expect certain statistical properties to emerge, but the exact details would depend on the specific characteristics of the arrangement.

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