independent events?

Question

Two coins are placed in a bag. One of the them is a fair coin, while the other is a special coin with tails on both sides. One of the coins is picked from the bag at random and this coin is tossed 2 times. Let Ai(i=1,2) be the event that the coin comes up tail on the i-th toss.

A1 and A2 seem be independent, but the calculation shows 
P(A2|A1) ≠P(A2) P(A1)

My question is why it is the case, are there any methods to know that they are independent without calculation?

匿名 · Accepted Answer

that's easy! do it!

? · Answer

If the picked coin is the biased one, than A1 and A2 have the same result.
A1 and A2 are related.

I don't get it. How come?
If the picked coin is the biased one, then A1 and A2 have the same result.
Why does this show A1 and A2 are related?

2014-10-15 16:33:34 補充：
A1 和 A2 （在 biased 的情況下）必為相同，即兩者不獨立。（於本情況）
So, if the biased coin is not 2-face such as the biased coin comes up tail with 2/3, are A1 and A2 independent?

2014-10-15 20:15:59 補充：
OK, Thanks.

知足常樂 · Answer

你應該想講 P(A2 ∩ A1) ≠ P(A2) P(A1) 吧？

A and B are independent
iff Pr(A ∩ B) = Pr(A) Pr(B)
iff Pr(A | B) = Pr(A)
iff Pr(B | A) = Pr(B)

2014-10-15 12:38:02 補充：
你問的東西不難解答。

A1 和 A2 明顯是 dependent 的。

If the picked coin is the biased one, than A1 and A2 have the same result.
A1 and A2 are related.

This is the logic without showing mathematically.

2014-10-15 15:58:29 補充：
你要求一個 non-calculating approach 去想，所以我完全不用公式。

若 A1 和 A2 為 independent events，那麼一件事的發生不會影響另一件事的發生。

但 A1 和 A2 （在 biased 的情況下）必為相同，即兩者不獨立。（於本情況）
當然，其實你也要看 unbiased 的情況才知。

若你容許數學，那麼
Pr(A2|A1 and biased) = 1
但
Pr(A2) 不是 1

註：Ai 為第 i 次得 tail 的event

2014-10-15 15:59:34 補充：
當然，這裏只show到
A2 和 "A1 and biased" 不是 independent

所以要真正 show A1 和 A2 independent 都是 用 definition 和 數式比較好～

2014-10-15 19:45:12 補充：
也不是，那要再看詳細才知。
總之最好用數計。

以上可以不計因為明顯見到
Pr(A2|A1 and biased) = 1
但
Pr(A2) ≠ 1

即知 Pr(A2|A1 and biased) ≠ Pr(A2)

independent events?

3 個解答