Difference between mutually exclusive and independent?

Leaving Cert Student

I can't work this out and have only got vague answers from teachers

GreeBo

independent means that they are not related
mutually exclusive means you cant have both, its either one or the other.

The positions of a light switch are mutually exclusive, it cant be on and off at the same time
Your cooker and your light switch are independent, they dont interact with each other.

Leaving Cert Student

so in math terms? how would you differentiate (for want of a better word) between the two?

weirdspider

In mutually exclusive events no intersection exists between the 2 events.
In independant events, there can be an intersection. One just doesn't cause the outcome of the other.

GreeBo

well looking at probabilities for example

The probablility of meing being dead and it being a sunny day are two independent events.

The probability of me being dead and me being alive are two mutually exclusive events (FYI I am not a cat owned by Mr Schrodinger)

Leaving Cert Student

This helped a lot thanks!

Peregrine

Here it is in Venn Diagram form, if it helps with probability.

Mutually exclusive:



They cannot happen together, probability of both of them happening together is 0 therefore there is no intersection.

Independent:



They can happen separately or they can happen together. There can be an intersection. Like GreeBo said:

The probablility of me being dead and it being a sunny day are two independent events.

The are independent of each other and can happen at the same time. GreeBo could be dead or it could be sunny or GreeBo can be dead and it can still be sunny That sounds like I hate GreeBo

Leaving Cert Student

Thank you! really felt like the book wasn't clear enough for this one!

Brendan1234

I think the mathematical definition of the two is:
If A and B are mutually exclusive then, (A intersection = {0} (the null set)
If A and B are independent then,
P(A intersection = P(A) x P(B)

I know that it might look like its overcomplicating a pretty simple idea, but if it's asked in the exam, then that should get you the marks.

MathsManiac

Here's an example that might help.

Roll an ordinary unbiased die. Let E, F, G, H be the following events:
E: we get an even number
F: we get an odd number
G: we get a five or a six
H: we get a four, five or six.

E and F are mutually exclusive events. They cannot both happen. P(E∩F) = 0.

F and G are independent events, even though that might not be obvious. But if you check the probabilities you will see that they are: P(F) = 1/2. P(G) = 1/3. The event F ∩ G has only one element in it, namely getting a 5, so P(F∩G) = 1/6. The fact that this is equal to P(F) times P(G) tells you that these are independent.

F and H are not independent, however. P(F) = 1/2, P(H) = 1/2, but P(F∩H) is 1/6, which is not equal to P(F) times P(H). They are not mutually exclusive either, by the way.

If two events arise from things that are clearly unrelated, then it can be obvious that they are independent. For example, if you roll a die and toss a coin, it's pretty obvious that the event that you get a six on the die is independent of the event that you get a head with the coin.

If two events are related to the same activity, it can be less obvious. For example, if you draw two cards, one at a time, from a pack of cards, the event that the second card is a diamond is NOT independent of the event that the first is a club, while the event that the second is a queen IS independent of the event that the first is a club.

MathsManiac

Brendan1234 wrote: »

I think the mathematical definition of the two is:
If A and B are mutually exclusive then, (A intersection = {0} (the null set)
If A and B are independent then,
P(A intersection = P(A) x P(B)

I know that I might look like its overcomplicating a pretty simple idea, but if it's asked in the exam, then that should get you the marks.

This is correct, except that {0} is not the correct way to write the null set. You can write it as {} or using the symbol ϕ. (Circle with a line through it.)
If you write {0}, this is the set with one element, namely the number 0.