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These are two example where it seems they are both independent but the second one is not. Why?

First problem.

There are 100 balls numbered 1 through 100 in a bag and one ball is randomly picked and returned to the bag for the next pick.

There is event A that the ball picked is divisible by 2 and event B that the ball picked is divisible by 5. It is given that both are independent event which seems very much obvious because they cannot affect the next pick, since the ball is placed back.

Also P(A)=50/100 = 1/2, P(B)=20/100 = 1/5 and P(A ∩ B) = 10/100=1/10

i.e P(A ∩ B)=P(A).P(B)

I too, do not understand how P(A ∩ B) decide independent event?

Second problem.

Only difference, is that, the bag now contain 65 ball.

And now these event A and B become non-independent event. How?

I know now,

P(A ∩ B) ≠ P(A).P(B).

Again as I stated above, how the above equation, is able to determine independent event.

I know P(A ∩ B) = P(A).P(B) . But how does this determine independent event?

Thanks.

[edit:]

The exact phrase from the question.

First problem.

There are 100 tickets in a bag numbered 1 through 100 and a ticket is picked at random. Let A be the event that the number on the ticket is divisible by 2 and let B be the event that number on the ticket is divisible by 5. Show that A and B are independent.

Second problem.

Let A and B be defined as in First problem above where a ticket is picked at random from a bag containing 65 tickets numbered 1 through 65. In this case A and B are not independent.

First problem.

There are 100 balls numbered 1 through 100 in a bag and one ball is randomly picked and returned to the bag for the next pick.

There is event A that the ball picked is divisible by 2 and event B that the ball picked is divisible by 5. It is given that both are independent event which seems very much obvious because they cannot affect the next pick, since the ball is placed back.

Also P(A)=50/100 = 1/2, P(B)=20/100 = 1/5 and P(A ∩ B) = 10/100=1/10

i.e P(A ∩ B)=P(A).P(B)

I too, do not understand how P(A ∩ B) decide independent event?

Second problem.

Only difference, is that, the bag now contain 65 ball.

And now these event A and B become non-independent event. How?

I know now,

P(A ∩ B) ≠ P(A).P(B).

Again as I stated above, how the above equation, is able to determine independent event.

I know P(A ∩ B) = P(A).P(B) . But how does this determine independent event?

**Here too in the second problem, both A and B seems to be, not affected by each other, just the same way, they seem obviously independent event as in the first problem. But yet, they are now non-independent event.**Thanks.

[edit:]

The exact phrase from the question.

First problem.

There are 100 tickets in a bag numbered 1 through 100 and a ticket is picked at random. Let A be the event that the number on the ticket is divisible by 2 and let B be the event that number on the ticket is divisible by 5. Show that A and B are independent.

Second problem.

Let A and B be defined as in First problem above where a ticket is picked at random from a bag containing 65 tickets numbered 1 through 65. In this case A and B are not independent.

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