# Are these three events independent?

In a given scenario - 1. Suppose that 9% of all employees in a given company are locals, 14% are men, and 18% have graduate degrees from top 10 schools. Let A, B and C be, respectively, the events that a randomly selected individual from this population is local, a man, and has graduated from top 10 school.

1) Do you believe that A, B and C are independent events? Explain why or why not.

2) Assuming A, B and C are independent events, find the probability that a randomly selected manager from this company is a non-local female and has earned an top-10 graduate business school. (only consider two ethnicities here- local and non-local, and two genders- male and female)

For question 1- I think that these are dependent as it is likely that out of 18% top ten group few could be men (given 14% are men), but I am not 100% sure. Can anyone pitch in and clarify? Also I cannot conclude if A is independent or not. Thanks!

• This does not seem like a question that has a definite answer given the information you've provided. Is this all the information we have in this problem? Also, if this is a homework or textbook question, could you please add the [self-study] tag? – Ruben van Bergen May 10 '17 at 11:25
• Yes this is from a book on Data Analysis and Decision Making, but not homework. This is the entire question as it is from the book. – B_perl May 10 '17 at 11:39

The dependence/independence of the the events depend on how many of the top ten group are men..

If $B$ & $C$ are independent then $P(B/C)= P(B)$

That is, probability of an employee is a man should not be affected by him being among the top ten group or not. If you observe 14% of employees among the top ten group are men then you can say B & C are independent. You need to look at $A \cap B$, $B \cap C$ and $A \cap C$ to decide the the dependence/independence of the events.

Update

1) Can not ascertain the dependence or independence from the data.

2) Assuming independence, $P( A^c \cap B^c \cap C) = P(A^c) * P(B^c) * P(C) = 0.91 * 0.86 * 0.18 = 0.14$

• Thanks @ab90hi! I do not have any other information other than I provided above, so thats why I am unable to conclude about the independence. I am posting the other parts of question as well. – B_perl May 10 '17 at 11:41
• Also when you say " If you observe 14% of employees among the top ten group are men then you can say B & C are independent." Would it not make it dependent instead of independent? – B_perl May 10 '17 at 12:03
• @B_perl No. I think you are getting confused with the Independence concept. Independence states that event C has no effect probability of B. – ab90hi May 10 '17 at 12:34
• Thanks a lot for correcting! so now for the given data, I understand that we cannot decide if these variables are dependent or independent. – B_perl May 10 '17 at 12:43
• @B_perl Assume that none of the men are graduates from top ten colleges then do you think B & C are independent? – ab90hi May 10 '17 at 12:43