# I cannot understand what the authors want to say exactly. ("Introduction to Algorithms 3rd Edition" by CLRS. Keywords: Two events are independent)

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?

• 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. Jun 12, 2021 at 14:52
• @littleO Thank you very much for your advice! Jun 12, 2021 at 14:53