A manufacturer of computer keyboards has assembly plants at two different locations, one in Iowa and the other in South Carolina. The keyboard faults in general are classified into three different categories depending on the location: fault related to a letter key, a number key, or other function key. The summary of last year’s inspection of production estimated the fault rates as follows:
If a keyboard from this manufacturer is chosen at random, are the events “faulty letter key” and “produced in South Carolina facility” independent of each other?
I just want to ask and verify if my reasoning is correct. First of all, to check if two event are independent, you simply use $P(A \mid B) = P(A) P(B)$ (where $A$ and $B$ are the events you want to check for independence. If you multiply those things, and when they are equal, then they are independent.
My $P(A)$ is the fauly letter key and its $90/250$ and $P(B)$ (from South Carolina) is $150/250$. Are those numbers correct? I just simply count the number it matches in the table over the total items. How about $P(A \mid B)$? How do we get it? Thanks.