Probability | related to combinations

[pwoz]

The question is as follows:
Suppose a lot of 50 electrical components contains five components that are defective. The lot is randomly divided among five customers, with each customer receiving 10 components.
Calculate the probabilities of the following events:
a. Each customer receives one defective component.
b. One customer receives all five defective components.
c. Two customers receive two defective components, one receives one, and the other two receive no defective components.


I'm looking for help as to why N(S) and N(X) are constructed the way they are.

I constructed N(S) as follows
[PLAIN]http://img203.imageshack.us/img203/2478/20647565.jpg

My understanding is the 5! in the denominator is because the order that the widgets are assigned does not matter. ie, the "first" person going first is the same as the "second" person going first.

To this end I constructed the formula for part A as:
[PLAIN]http://img27.imageshack.us/img27/4980/87152546.jpg

My logic on the coefficient is the same as before. It does not matter who goes first.

Part b was constructed as:
[PLAIN]http://img683.imageshack.us/img683/8265/11889894.jpg

I set the coefficient this way since there are 5 ways to pick the person who gets all of the defective widgets, and it doesn't matter who.

Part c:
[PLAIN]http://img265.imageshack.us/img265/7426/51203459.jpg

Again, I set the coefficient up as 5 ways to pick the first person, 4 for the second, and 3 for the last person to receive one.

I'm fairly certain I am not thinking about the coefficient term correctly. I'm not looking for a solution to these problems, rather I'm trying to understand the setup. Something about this topic has never clicked with me. Any guidance is appreciated.

 

[nate808]

Thank you for your question and for sharing your thought process with us. It is great to see that you are actively trying to understand the concepts and underlying logic behind the calculations.

To answer your question, let's first define the terms N(S) and N(X). N(S) represents the total number of possible outcomes in a given scenario, while N(X) represents the number of outcomes that satisfy a specific condition X. In other words, N(X) is the number of successful outcomes out of all possible outcomes.

In the given scenario, N(S) is the total number of ways in which 50 components can be divided among 5 customers, without any restrictions. This can be calculated using the combination formula, which you have correctly used as (50 choose 10) * (40 choose 10) * (30 choose 10) * (20 choose 10) * (10 choose 10). This gives us a total of 2.1187e+13 possible outcomes.

Now, let's look at the specific events mentioned in the question and how we can calculate N(X) for each of them.

a. Each customer receives one defective component:
For this event, we need to calculate the number of ways in which we can choose 5 defective components out of a total of 50 components. This can be calculated using the combination formula as (50 choose 5). However, this counts all possible combinations of 5 defective components, without considering the distribution among the customers. Since we want each customer to receive only one defective component, we need to divide this by the number of ways in which 5 components can be distributed among 5 customers, which is 5!. This gives us N(X) as (50 choose 5) / 5! = 2.1187e+13 / 120 = 1.7656e+11 successful outcomes.

b. One customer receives all five defective components:
For this event, we need to calculate the number of ways in which we can choose 5 defective components out of a total of 50 components. However, this time we want all 5 components to be assigned to one specific customer, so we need to multiply this by 5 (since there are 5 customers to choose from). This gives us N(X) as 5 * (50 choose 5) = 5 * 2.1187e+13 / 120 = 8.