I have a coin that lands on heads with probability $N$ and tails with probability $1-N$. How do I explain the outcomes of the successive flips of the coin are independent of each other, knowing only $N$. Also if I didn't know the value of $N$, are the outcomes of the successive flips of the coin independent of each other?
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2$\begingroup$ That should be "lands heads with probability $N$ and tails with probability $1-N$", yes? $\endgroup$ – David Feb 18 '14 at 23:14
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$\begingroup$ comment now redundant $\endgroup$ – oks Feb 18 '14 at 23:14
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$\begingroup$ @David Yes, corrected my question now. Thanks. $\endgroup$ – orange Feb 18 '14 at 23:15
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$\begingroup$ still $N-1$ instead of $1-N$ $\endgroup$ – Denis Feb 18 '14 at 23:23
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2$\begingroup$ It's rather unusual to use $N$ for denoting a fractional number. Apart from that, I don't understand the bit about "explain the successive flips of the coin are independent of each other" $\endgroup$ – leonbloy Feb 18 '14 at 23:33
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I'm taking that each coin flip has $\mathbb{P}(\text{heads}) = N$ and $\mathbb{P}(\text{tails}) = 1-N$.
If two events A, B are independent, then $\mathbb{P}(A) = \mathbb{P}(A|B)$. Let $A$, $B$ be two successive coin flips. Then we can see that the coin flips must be independent given the above information.
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$\begingroup$ the concept associate with the ℙ(A)=ℙ(A|B) is called marginal independence. $\endgroup$ – Juan Zamora Apr 18 '18 at 17:07
