We model the experiment where we toss a fair coin n times as follows. The sample space ? is the space of (ordered) binary vectors of length n. A 1 in the i-th position of such a binary vector indicates Heads at the i-th toss. The probability measure P on ? is defined as P(A)=|A|/|?| for all A??. Let Ai be the event that the ith toss is Heads, i = 1,...,n.

Prove, using the definition of independence, that events A1, . . . , An are mutually independent events.

We model the experiment where we toss a fair coin n times as follows. The sample space 2 is the space of (ordered) binary vectors of length n. A 1 in the i-th position of such a binary vector indicates Heads at the i-th toss. The probability measure P on 2 is defined as P(A) = [A| /|2) for all A C 2. Let A; be the event that the ith toss is Heads, i = 1, ..., n. Prove, using the definition of independence, that events Al, ..., An are mutually independent events