3
$\begingroup$

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 
  • 6
    "Realistically" is not a mathematical question.2012-10-09
  • 2
    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.2012-10-09
  • 0
    @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 6.2012-10-09
  • 0
    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?2012-10-09
  • 1
    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.2012-10-09
  • 0
    Michael Hardy and I interpreted the question differently. He looked for a supergroup where the probabilities would be the same regardless of the new data. I looked for a subgroup where the selection of one individual and being told one quality gave no information on the other quality. His approach has the advantage of giving a simple answer, so is probably the correct reading.2012-10-10

3 Answers 3