Confusion about independence and dependence

[KataKoniK]

I got a mid-term coming up and I am a bit confused about knowing which events are independent or dependent. I know that if you're drawing balls for example without replacement, then it is dependent otherwise it is independent.

My question is, just say you are given the following events:

A = The first roll is a six
B = The last roll is a six
C = The first and last rolls are the same
D = The sum of ten rolls is eleven or less

Are events A and B independent?

Are events A and C independent?

Are events A and D independent?

If all the answers are yes, can someone explain why? I am a bit confused with the concept here.

 

[BicycleTree]

Two events are independent if knowing one of them means you can't say anything additional about the probability of the other one of them. A and B, and A and C, are independent, but A and D are not independent because if the first roll is a 6 it means that "the sum of ten rolls is eleven or less" is much less likely to happen (in fact it cannot happen).

 

[TenaliRaman]

A mathematical way to confirm this is,
show either,
P(X/Y) = P(X)
or
P(X n Y) = P(X)P(Y/X) or P(Y)P(X/Y)

if you can show any of the above as true then you are done.

For now just consider that only two rolls were made.
P(A) = 1/6
P(B) = 1/6
P(A n B) = 1/36
P(A)*P(B) = 1/36
so A and B are independent

P(A) = 1/6
P(C) = 6/36
P(C/A) = 1/6 = P(C)
So A and C are independent

This can be easily extended for ten rolls as well.

-- AI

 

[KataKoniK]

Thank you. I got another question I forgot to include in my original post. If a situation involves picking out items from a box without replacement, does it mean that the events are mutually exclusive? And if you're picking stuff out of the box with replacement, then it's not mutually exclusive? Or?

 

[HallsofIvy]

Thank you. I got another question I forgot to include in my original post. If a situation involves picking out items from a box without replacement, does it mean that the events are mutually exclusive? And if you're picking stuff out of the box with replacement, then it's not mutually exclusive? Or?

What's "not mutually exclusive"?? Be very careful how you words statements or questions (as I've said before, mathematics is very precise).

"Mutually exclusive" means that two events CANNOT both happen- one happening excludes the other. Whether two events are or are not mutually exclusive depends upon what the events are!

For example, suppose the "items" in your box are slips of paper numbered 1 to 100. You pick a slip of paper from the box, put it in your pocket (do not replace it in the box), and then draw a second paper from the box. The events "the first paper had the number 78 on it" and "the second paper had the number 85 on it" are not mutually exclusive but the events "the first paper had the number 78 on it" and "the second paper had the number 78 on it" are mutally exclusive. If the first paper has the number 78 on it, that number is no longer in the box and can't be drawn the second time. With replacement, this would not be mutually exclusive since you could draw "78" both the first and second times.
On the other hand, whether you replace or not, the two events "the first paper has the number 78 on it" and "the first paper had the number 85 on it" are mutually exclusive.

Be very, very precise!

 

[KataKoniK]

I think I got it thanks, so is this right?

There's a box with 4 black, 6 red, and 10 white balls. Balls are drawn with replacement. You draw four times and if you draw a black ball, you win money (lose money otherwise). Since you draw four times, then there will be cases where you draw 0, 1, 2, 3, 4 black balls. So probability of those would be 16/20, 4/20, 4/20, 4/20, 4/20 respectively?