# Probability

Suppose you have a true random bit generator where each bit in the generated stream has the same probability of being a 0 or 1 as any other bit in the stream and that the bits are not correlated, i.e., the bits are generated from identical independent distribution. However, the bit stream is biased. The probability of a 1 is 0.5 + d and the probability of a 0 is 0.5 – d where 0 <d < 0.5. A simple deskewing algorithm is as follows: Examine the bit stream as a sequence of nonoverlapping pairs. Discard all 00 and 11 pairs. Replace each 01 pair with 0 and each 10 pair with 1.

a. What is the probability of occurrence of each pair in the original sequence?

b. What is the probability of occurrence of 0 and 1 in the modified sequence?

c. What is the expected number of input bits to produce x output bits?

d. Suppose that the algorithm uses overlapping successive bit pairs instead of nonoverlapping successive bit pairs. That is, the first output bit is based on input bits 1 and 2, the second output bit is based on input bits 2 and 3, and so on. What do you infer about the output bit stream?

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