I need someone to explain why my method doesn't work (based on my Venn diagram)

burgerandcheese

burgerandcheese

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"In a class of 25 students, 10 are members of the Mathematics Club, 18 are members of the Science Club, 5 are members of both clubs. A student is picked at random from the class. Find the probability that the student is a member

(a) of the Mathematics Club or the Science Club.
(b) of the Mathematics Club and the Science Club.
(c) of the Mathematics Club only.
(d) of the Science Club only."

I just need help for part (b)
I drew a Venn diagram and I had 13 in the set for Science Club, 5 in the set for Mathematics Club, 5 in the intersection point of the two sets, and 2 outside of the sets.

I understand that the answer is 5/15 by simply looking at the diagram, but how come I don't get the same answer if I use the multiplication rule? 10/25 * 18/25 gives 36/125. I just don't understand why it can't be like that? Is it because they are dependent events?
 
burgerandcheese said:
"In a class of 25 students, 10 are members of the Mathematics Club, 18 are members of the Science Club, 5 are members of both clubs. A student is picked at random from the class. Find the probability that the student is a member

(a) of the Mathematics Club or the Science Club.
(b) of the Mathematics Club and the Science Club.
(c) of the Mathematics Club only.
(d) of the Science Club only."

I just need help for part (b)
I drew a Venn diagram and I had 13 in the set for only the Science Club, 5 in the set for only the Mathematics Club, 5 in the intersection point of the two sets, and 2 outside of the sets.

I understand that the answer is 5/15 by simply looking at the diagram, but how come I don't get the same answer if I use the multiplication rule? 10/25 * 18/25 gives 36/125. I just don't understand why it can't be like that? Is it because they are dependent events?
Click to expand...

Please note my corrections above. There are 18 members of the Science Club, not 13. The two sets in the Venn diagram are the Science Club and the Mathematics Club; the numbers you gave are for particular regions of the diagram, not for the sets the diagram is about.

The probability of being a member of the Mathematics Club and the Science Club is the number of members of both clubs, over the total number of students. You are told that the total number of students is 25, not 15! Was that just a typo?

The multiplication rule applies only to independent events, which is a very special situation, not something to be expected. You can't apply it here. Never apply it unless you know that events are independent.
 
Dr.Peterson said:
Please note my corrections above. There are 18 members of the Science Club, not 13. The two sets in the Venn diagram are the Science Club and the Mathematics Club; the numbers you gave are for particular regions of the diagram, not for the sets the diagram is about.

The probability of being a member of the Mathematics Club and the Science Club is the number of members of both clubs, over the total number of students. You are told that the total number of students is 25, not 15! Was that just a typo?

The multiplication rule applies only to independent events, which is a very special situation, not something to be expected. You can't apply it here. Never apply it unless you know that events are independent.
Click to expand...

Yes sorry I was struggling on how to word it properly, and yes it was a typo. I meant 5/25.

How do I know that these two events are not independent?
 
burgerandcheese said:
Yes sorry I was struggling on how to word it properly, and yes it was a typo. I meant 5/25.

How do I know that these two events are not independent?
Click to expand...

Because the product rule didn't work! That can be taken as a definition of independence (or at least equivalent to the definition).

This is why I said: Never apply it unless you know that events are independent. The default assumption is that they are not. You don't ask "How do I know that they are not independent?"; you ask, "Is there any reason to think that they are?" In the absence of a reason, assume not.
 
Dr.Peterson said:
Because the product rule didn't work! That can be taken as a definition of independence (or at least equivalent to the definition).

This is why I said: Never apply it unless you know that events are independent. The default assumption is that they are not. You don't ask "How do I know that they are not independent?"; you ask, "Is there any reason to think that they are?" In the absence of a reason, assume not.
Click to expand...

OK! I will keep that in mind :rolleyes:
 
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