The following formulas are also provided to save you the trouble of looking them up elsewhere: . Binomial coefficient n! K = k!(n - k)! . Inclusion-exclusion principle P(AUB) = P(A) + P(B) - P(An B) . Chain rule P(AnB) = P(A|B) P(B) . Odds P(A) P(A) Odds(A) = P(A) ~ 1- P(A) . Conditional probability P(A B) = P(An B) P(B) . Bayes' theorem P(E H ) P(Hi) P(HE)= P(E Ho) P(Ho) + P(E|HI) P(H1) + . .. + P(EH,.) P(H,) . Information content I(A) = - 10g2 P(A) = log2(1/P(A)) . Entropy H(X) = -> P(x;) log2 P(xi) i= 11. (18 points) Below are two alphabets, each consisting of five symbols with varying probabilities. Alphabet 1 Alphabet 2 Symbol a b C d e Symbol a b C d e Frequency 0.2 0.1 0.1 0.2 0.4 Frequency 0.1 0.2 0.05 0.01 0.64 Below are two symbol codes for encoding the symbols a, b, c, d, e. Code A Code B Symbol a b C d e Symbol a b C d e Code word 101 11 1001 1000 0 Code word 11 101 100 01 00 (a) Among the symbols of both alphabets, which symbol in which alphabet has the greatest information content? What is that information content? (b) Find the entropy of alphabet 2. (c) Each of these two codes is a Huffman code for one of the two alphabets. Which code belongs to which alphabet? How do you know