How do I know when to use a Venn diagram or a probability tree? Also, when can I assume that the events are independent?
Thank you for asking this question. It helped me realise the following ! :
There is no strict rule for using either one or other technique. You need to try and model the question in both ways and more (probability tables, equations etc). The model that can satisfy the question best with, is good for you, for that question. In fact mostly the question is worded in a way where it encourages you to use a certain model.
Do not assume anything unless it is specifically mentioned. Just try to model the information in question with the techniques you know.
For question 1 I felt "Tree diagram" is the best way to model the problem. And for second one I felt more comfortable with Venn diagrams and then algebraic equations to solve for unknowns. I tried but could not create a satisfactory Venn model of first problem.
In general I find Tree diagrams easy to deal with in conditional probability and Venn diagrams in sets or very simple probability questions....but maybe that is just me !
Question 1 (Tree would be as follows)
[1]-----0.38----F-----0.13---L
[2]-----0.38----F-----0.87---R
[3]-----0.62----M-----0.24---L
[4]-----0.62----M-----0.76---R
P(L) = [1]+[4] = 0.13x0.38+0.24x0.62
P(F|L)=[1]/([1]+[3]) = (0.13x0.38)/((0.13x0.38)+(0.24x0.62))
Question 2
x = Only R not B
y = R and B
z= Only B not R
x+y = 0.71
x+z = 0.44
1-y-z = 0.21
solving i get x = 0.18, y=0.53,z=0.26, and that gives you the answer to the second problem.
So your first issue is that your Venn diagram does not display the given information, as you note at the end. This is confusing you, despite your solution being essentially correct, if somewhat oddly calculated (though I have no idea why you're trying to calculate $9 - 16$: you don't need to adjust the outer values, because those are already in your Venn diagram correctly; indeed, they're given in the question).
Let's call the number of students who like all three subjects $x$. Then your actual Venn diagram looks like this:
Now, summing all of those values, we see that $80 = 24 + 9 + 16 + 9 + 7 - x + 9 - x + 12 - x + x = 86 - 2x$, and so $2x = 6$, and $x = 3$. Thus, the full Venn diagram looks like this:
We can now solve the questions by just reading off the diagram.
Best Answer
You should use the fact that there are 39 people total, and that every person is in at least one club.
This means that if you add up the numbers in your Venn Diagram, you should get 39. Can you solve the rest of the problem from here?