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I got this question in my hw practice set

In a class, there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?

I solved the number (i believe is 9) that make class and gender independent. But it got me thinking: under what scenario, can class and gender be dependent? (i mean, class and gender are totally unrelated things right? therefore, they should be independent correct?)

I tried cook up some numbers to show that they're dependent(see table below). Mathematically, I've shown, through the following table, that gender and classification are indeed dependent. But how can gender and class classification be dependent?

            Male Female Total
Freshman     18    20    38
Sophomore    12    16    28          
Total        30    36    66
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closed as off topic by Michael Greinecker, David Mitra, tomasz, Noah Snyder, no identity Oct 10 '12 at 11:42

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6  
"Realistically" is not a mathematical question. –  Qiaochu Yuan Oct 9 '12 at 18:08
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For independence, we want the gender to give us no information about the class, and vice-versa. Before you added the sophomore girls, if person is picked at random and turns out to be a girl, you know for sure that the person is a freshperson. –  André Nicolas Oct 9 '12 at 18:13
    
@QiaochuYuan Suppose that we toss 2 fair dice. Let E1 denote the event that the sum of the dice is 6 and F denote the event that the first die equals 4. Hence, E1 and F are not independent. Realistically, the reason for this is clear because if we are interested in the possibility of throwing a 6 (with 2 dice), we shall be quite happy if the first die lands on 4 (or, indeed, on any of the numbers 1, 2, 3, 4, and 5), for then we shall still have a possibility of getting a total of 6. –  user133466 Oct 9 '12 at 18:17
    
If, however, the first die landed on 6, we would be unhappy because we would no longer have a chance of getting a total of 6. In other words, our chance of getting a total of 6 depends on the outcome of the first die; thus, E1 and F cannot be independent. So is there an explanation like this for class and gender? –  user133466 Oct 9 '12 at 20:09
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Until recently, people thought cervical cancer and human papilloma virus were "totally unrelated", but they were wrong. It turns out one causes the other. In this problem, you do observe a relationship between class and gender; if you want to know why this relationship exists, you have to investigate further. Maybe an evil witch abducted all female babies born in 1996, which is why there are no sophomore girls this year. –  Nate Eldredge Oct 9 '12 at 20:49

3 Answers 3

One answer to "how can they be dependent" is simply that the number of sophomore girls is something other than $9$. In particular, suppose all the sophomores were boys, whereas among freshmen there were some boys and some girls. Then the conditional probability that one of these is a boy, given that the person is a sophomore, would differ from the conditional probability that the person is a boy, given that he or she is a freshman. So there's a lack of dependence.

Dependence and independence are here taken to mean dependence and independence only within this small group, not within any larger population from which one might suppose the small group to be a random sample. Even if gender and class are independend in the population as a whole, one could get a small sample in which they are dependent.

Now suppose this were in a population among whom it is customary that those of one gender drop out of school after their freshman year. There you'd have dependence.

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why would you think of this in terms of conditional probability? I'm thinking it's simple union and interects –  user133466 Oct 10 '12 at 1:25
    
@user133466 : You can phrase the whole thing in that language too. It's a trivial difference. –  Michael Hardy Oct 10 '12 at 2:21
    
oh no =( this is so convoluted. I started a post before on how I can identify a probability problem as a conditional probability problem(math.stackexchange.com/questions/205103/…). it seems like you're suggesting they're the same. this is still confusing =( –  user133466 Oct 10 '12 at 4:24

This has a precise mathematical meaning. There are four "events": S=sophomore, F=freshman, G=girl, B=boy. All are parts of your "universe". You can construct new events, for instance F&G would mean the intersection of F and G, that is "freshman girls". Now you can compute the frequency of each event, say p(S&B) would be 6/N because there are 6 sophomore boys in a universe of N people (unfortunatelly N is unknown). Now, two events like "G=girl" and "F=freshman" are said to be "independent" if p(G&F) equals the product p(G).p(F) (why? later...) Let us compute. If the number of sophomore girls is "x", the total population will be 4+6+6+x=16+x. So independence will imply 6/(16+x)=(6+x)/(16+x).10/(16+x). The solution is x=9 and you are right. Any other solution would mean that the events are not independent. Also you could check that the solution does not change when you try with p(G&S) and so on...

The notion of independence is cooked to obtain that the frequency of "freshman girls" (G&F) among "freshman people" (F), that is 6/10, or the frequency of G&S among S, that is 9/15, both equal the total frequency of girls in the universe, that is 15/25. The same thing with boys. If this happens, clearly "class" does not depend on "gender".

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The only thing you can't do is have both some freshman girls and some sophomore boys in the mix. They are dependent if you select one individual from your group, are told the class, and have more information about the gender. So if you had three freshman girls and three sophomore boys, were told that a freshman was picked, you would know she was a girl. Before being told the class, it was 50/50. This is a dependent case.

If you have only sophomores, then being told the class doesn't help with the gender. Similarly, being told the gender doesn't help with the class (which you already know).

If your group has both freshman girls and sophomore boys they are not indpendent.

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Ross, thank you for your answer! I upvoted everyone who contributed, honestly I still don't quite get it. but it's pretty late tonight. I'll think more on each answer when i wake up tomorrow morning. –  user133466 Oct 10 '12 at 4:26
    
The important thing is whether learning one answer (class) changes your estimate on the odds on the other criterion (gender). If the proportion of males among the freshmen is different from the sophomore, and you were told somebody was a freshman, you would revise your estimate of the chance that s/he was male. Michael Hardy's model has each class 3:2 female:male, so you learn nothing from the class that helps with gender. This is the point of the exercise. –  Ross Millikan Oct 10 '12 at 4:31

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