A biased coin that comes up heads with probability p is flipped n times. Let $A_i$ denote the event that the $ith$ flip results in heads (i = 1...n)
a) Calculate the probability of $\bigcup_i^nA_i$ by applying De Morgan's Law and then independence of the $A_i$'s
b) Calculate the probability of $\bigcup_i^nA_i$ by applying the inclusion-exclusion principle first then independence of the $A_i$'s
c) By comparing results of the previous parts, prove the identity
$(1-p)^n$ = $\sum_{k=0}^n {n \choose k} (-1)^kp^k$
for any n > 0 and $0\le p\le1$ real number.
d) From the above, prove
$(a-b)^n$ = $\sum_{k=0}^n {n \choose k} (a)^{n-k} (-b)^k$
for arbitrary positive reals a and b
PROGRESS:
Ok, so I have for part a) that $\bigcup_i^nA_i$ by De Morgan is equivalent to $(\bigcap_i^nA_i^c)^c$ then by independence we have the probability is $1-(1-p)^n$
For part b) the answer is clearly going to be equivalent to part a) but written in a different form but I am not sure where to go from there. Any help much appreciated!