dependent or independent

Hello!
So, I've got a difficulty understanding whether 2 events are dependent or independent.
There is a task:
On a given day, photocopier A has a 10% chance of malfunctioning and machine B has a 7% of the same. Given that at least one of the machines malfunctioned today, what is the chance that machine B malfunctioned?
My solution.
So, I guess the task is really simple. event A - machine A malfunctioned, event B - machine B malfunctioned We just need to calculate P(B | AuB ) = P (B n (AuB) ) / P (AuB). P(B n (AuB)) should be equal to P(B) (can be easily deduced from the venn diagram). But P(AuB) = P(A) + P(B) - P(AnB). And here I assume that events A and B are independent, so P(AnB) = P(A)*P(B). And the result turn out to be correct = 42.9% (checked the answers).
But in the next task.
On any day, the Probability that a boy eats his lunch is 0,5. The probability that his sisters eats her lunch is 0.6. The probability that the girl eats her lunch given that the boy eats his is 0.9.
I won't write here what I need to find, the task is quite simple. What's interesting is that here it is stated that two events are dependent. That is P(sister eats) != P(sister eats | brother ate). But how can I know then that the 2 events in the previous task are independent. Perhaps, they're also dependent, as nothing was said about it. How do you distinguish independent and dependent events without using the rule P(A) = P(A|B) iff events are independent. Thanks for the answer in advance

silsaty said:

Hello!
So, I've got a difficulty understanding whether 2 events are dependent or independent.
There is a task:
On a given day, photocopier A has a 10% chance of malfunctioning and machine B has a 7% of the same. Given that at least one of the machines malfunctioned today, what is the chance that machine B malfunctioned?

Click to expand...

This is impossible to answer without knowing whether or not the two probabilities are independent! I suppose it is possible to imagine that they are dependent- if the two machines are right next to one another, machine A, malfunctioning and overheating might well cause machine B to overheat and so malfunction! But, in that case, in order to answer a question like this you would have to be told how one machine malfunctioning depends upon the other. Since you are NOT told that, you have no choice but to assume the are independent.

My solution.
So, I guess the task is really simple. event A - machine A malfunctioned, event B - machine B malfunctioned We just need to calculate P(B | AuB ) = P (B n (AuB) ) / P (AuB). P(B n (AuB)) should be equal to P(B) (can be easily deduced from the venn diagram). But P(AuB) = P(A) + P(B) - P(AnB). And here I assume that events A and B are independent, so P(AnB) = P(A)*P(B). And the result turn out to be correct = 42.9% (checked the answers).

Click to expand...

Here is what I would do: imagine 1000 "events". In 1000(.10)= 100 them, machine A malfunctions. In 1000(.07)= 70 of them, machine B malfunctions, and in 1000(.10)(.07)= 100(.07)= 7 of them both malfunction. If we just add 100+ 70= 170, we would be counting those events in which both A and B malfunctioned twice- so subtract those off. That is, there are 100+ 70- 7= 163 incidents in which A or B or both malfunction. Of those 163 events, in which "at least one machine malfunctioned", 70 were events in which B malfunctioned. Given that at least one malfunctioned, the probability that B malfunctioned is 70/163= 0.429..., exactly what you have.

But in the next task.
On any day, the Probability that a boy eats his lunch is 0,5. The probability that his sisters eats her lunch is 0.6. The probability that the girl eats her lunch given that the boy eats his is 0.9.
I won't write here what I need to find, the task is quite simple. What's interesting is that here it is stated that two events are dependent. That is P(sister eats) != P(sister eats | brother ate). But how can I know then that the 2 events in the previous task are independent. Perhaps, they're also dependent, as nothing was said about it. How do you distinguish independent and dependent events without using the rule P(A) = P(A|B) iff events are independent. Thanks for the answer in advance

Click to expand...

In the second problem you are told "the probability that the girl eats her lunch given that the boy eats his is 0.9". If the two probabilities are dependent you must be given the "contingent" probability or it is impossible to do the problem.