$($a$)$ Compute the bigram count table, C$($w$2|$w$1)$ for the sentence "I saw Susie sitting in a shoe shine shop. Where Susie sits Susie shines, and where Susie shines Susie sits." Put w$1$ in the left hand column, and w$2$ in the top row. Include punctuation, clitics, and sentence start and end markers as individual tokens, and index words using their lemmatized forms. $(1$ pt$)$

$($b$)$ Compute the bigram probability table, P$($w$2|$w$1)$ for above sentence, assuming the following overall unigram counts: C$($I$)=2,$ C$($see$)=25,$ C$($Susie$)=10,$ C$($sit$)=20,$ C$($in$)=80,$ C$($a$)=90,$ C$($shoe$)=50,$ C$($shine$)=20,$ C$($shop$)=20,$ C$($Where$)=60,$ C$(,)=80,$ C$($and$)=100.$ Assume there are $75$ sentences in the corpus, and they all end with a period. $(1$pt$)$

$($c$)$ Compute the probability and perplexity of the first sentence in $($a$)$ using the bigram approximation. $(1$pt$)$