A plagiarism detection service uses locality sensitive hashing $($LSH$)$ to find

similar documents. Suppose the database has $100,000$ documents that you need to analyze to find similar documents. You have the memory capacity to compute document signatures of length $1024,$ and you set the number of bands to be $64$ and the size of each band to be $16$ rows:

a$.$ What is the probability that two documents that are $80\%$ similar get assigned to the same bucket?

b$.$ What is the probability that two documents that are $25\%$ similar get assigned to the same bucket?

c$.$ What is the probability that two documents that are $50\%$ similar get assigned to

different buckets?