# Probability calculation of dependent events with limitations

• MHB
• Slyv3r
In summary: Your Name]In summary, the person posting on the forum is seeking help with calculating the probability of the production line halting after the 7th order is picked. They provide information about a shipping company for clothes that has 10 orders with 4 clothes each, and only 6 order locations. The expert summary explains that the probability can be calculated using the combination and permutation formulas, and the final probability is 0.2603 or 26.03%. The person also apologizes for any language errors and welcomes any further questions or information.
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.

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.

## 1. What is the definition of dependent events in probability?

Dependent events in probability are events that are influenced by or affected by the outcomes of previous events. This means that the probability of one event occurring is dependent on the outcome of another event.

## 2. How do you calculate the probability of dependent events?

To calculate the probability of dependent events, you need to multiply the probabilities of each event occurring. This is known as the multiplication rule. The formula for this is P(A and B) = P(A) * P(B|A), where P(A) is the probability of event A occurring and P(B|A) is the probability of event B occurring given that event A has already occurred.

## 3. What are some limitations when calculating the probability of dependent events?

One limitation when calculating the probability of dependent events is that the outcomes of the events must be independent. This means that the outcome of one event should not affect the outcome of the other event. Additionally, the events must be mutually exclusive, meaning that they cannot occur at the same time.

## 4. How do you handle dependent events with overlapping outcomes?

If the events have overlapping outcomes, you need to adjust the probability calculation to avoid counting the overlapping outcomes twice. This can be done by subtracting the probability of the overlapping outcomes from the total probability. For example, if event A and event B have an overlapping outcome, the formula would be P(A or B) = P(A) + P(B) - P(A and B).

## 5. Can you provide an example of dependent events with limitations?

One example of dependent events with limitations is drawing cards from a deck without replacement. If you draw a card from a deck of 52 cards, the probability of drawing an ace is 4/52. However, if you do not replace the card back into the deck and draw another card, the probability of drawing an ace again changes to 3/51 because there is one less ace and one less card in the deck. This shows how the outcome of the first event affects the probability of the second event.

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