Probability calculation of dependent events with limitations

[Slyv3r]

Dear forum,

First time posting and as English is not my native language, I'd like to apologize in advance for any linguistic errors I make.
Yesterday, I received a case which sounded really easy to calculate but for some reason I can't get my head around it.
This is the case:
In a shipping company for clothes there are 10 different orders being handled at the same time. These orders all have 4 clothes, which means there are 40 clothes in total that have to be sorted. At random these clothes are being picked from the stock and assigned to an order. The issue is, that there are only 6 order locations (packing and shipping), while there are 10 orders to be picked. But, when 4 clothes are assigned to 1 order location, the order is complete (packed and shipped) and will open up again for a new order order. How do I calculate the probability that the whole production line will halt since the 7th order is being picked while there are only 6 locations.
It seems to me that there is a calculation for this but I just can't grasp it at the moment. I really do apologize if this is a basic question and if my brain just doesn't want to co-operate but for now I really have no clue how to get this to work.

If you have any other questions regarding the case or think you have some information that might help me out. Please let me know.

Thank you in advance.

 

[jvicens]

Dear forum,

Thank you for sharing your case with us. I can understand your frustration with trying to calculate the probability in this situation. It is not a basic question and it requires careful consideration and analysis.

To begin with, we need to determine the total number of possible combinations that can occur when picking 4 clothes from a stock of 40. This can be calculated using the combination formula, nCr = n! / (r!(n-r)!), where n is the total number of items (40) and r is the number of items being picked (4). In this case, there are 91390 possible combinations.

Next, we need to consider the number of ways in which these 40 clothes can be distributed among the 10 orders. This can be calculated using the permutation formula, nPr = n! / (n-r)!, where n is the total number of items (40) and r is the number of items in each order (4). In this case, there are 23751 possible ways in which the 40 clothes can be distributed among the 10 orders.

Now, let's consider the probability of the production line halting after the 7th order is picked. This can be calculated by dividing the number of possible combinations (91390) by the number of possible ways in which the 40 clothes can be distributed among 6 orders (23751). This gives us a probability of 0.2603 or 26.03%.

I hope this helps you in your calculations. If you have any other questions or need further clarification, please let me know.