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I am reading "Introduction to Algorithms 3rd Edition" by CLRS.

For example, suppose that we flip two fair coins and that the outcomes are independent. Then the probability of two heads is $\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$. Now suppose that one event is that the first coin comes up heads and the other event is that the coins come up differently. Each of these events occurs with probability $\frac{1}{2}$, and the probability that both events occur is $\frac{1}{4}$; thus, according to the definition of independence, the events are independent - even though you might think that both events depend on the first coin.

The authors wrote "even though you might think that both events depend on the first coin".
I cannot understand what the authors want to say exactly.
Suppose that we flip two fair coins and that the outcomes are independent.
What is the definition of "an event $A$ depends on the first coin" in this case?

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    $\begingroup$ You could omit that phrase and just read onward if you'd like. The important thing is to understand that if $A$ is the event that the first coin comes up heads and $B$ is the event that the coins come up differently, then $A$ and $B$ are independent events. Some people might find that fact to be surprising, because the description of event $A$ mentions coin 1 and the description of event $B$ also mentions coin $1$. If the result doesn't seem surprising to you, I think that's ok. $\endgroup$
    – littleO
    Jun 12, 2021 at 14:52
  • $\begingroup$ @littleO Thank you very much for your advice! $\endgroup$
    – tchappy ha
    Jun 12, 2021 at 14:53

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These events are independent even though we can't split the random experiment into two parts, where one part tells us everything about the first event, and the other tells us everything about the second event. That is a cheap way to get independence: for example, if I flip two coins and roll three dice, any event about the coins is independent of any event about the dice.

This example shows this is not the only way to get independence. Both "the first coin comes up heads" and "the coin comes up differently" depend on the first coin in that to determine whether they occurred or not, you need to know how the first coin landed. But they're still independent.

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  • $\begingroup$ Misha Lavrov, Thank you very much for your answer! $\endgroup$
    – tchappy ha
    Jun 12, 2021 at 15:20

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