Question

suppose that 1% of (binary) bits received by your smartphone have errors. what is the probability that the 27th bit received by your smartphone is the first erroneous bit?

Answer 1

The probability that any given bit received by the smartphone is erroneous is 1%, or equivalently, the probability that any given bit is correct is 99%.

The probability that the 27th bit is the first erroneous bit means that the previous 26 bits were all correct. The probability of this happening is:

P(all previous 26 bits are correct) = (0.99)^26

Now, given that the previous 26 bits were all correct, the probability that the 27th bit is erroneous is simply 1% or 0.01.

Therefore, the overall probability that the 27th bit received by the smartphone is the first erroneous bit is:

P(27th bit is first erroneous bit) = P(all previous 26 bits are correct) x P(27th bit is erroneous | all previous 26 bits are correct)

= (0.99)^26 x 0.01

= 0.0081 or approximately 0.81%