Question

Of all bit sequences of length 8, an 8-bit sequence is selected at random.
Assuming that the probability of a bit being 0 is equal to that being 1, determine
the probability that the selected bit sequence starts with a 1 or ends with the two
bits 00.

Answer 1

From the question , we can see that the event is either getting a 8 digit number starting with 1 or ending with 00.

So , sample space is 2⁸ as there are 8 places and we can put either 1 or 0 there , so sample space is 256 or we can say we have 256 equally likely outcomes.

Now , according to event , first space is 1 so similarly we have 2⁷ equally likely outcomes with first space as 1 , so we have 128 outcomes.

We can even have two zeros at last , so for that we have 2⁶ possibilities or 64 outcomes.

So total outcomes for events are 128+64 = 192

Now there must be some numbers which start with 1 and end with 00 , so we must remove them once as there are added twice in our 192 outcomes we got . So, we have fixed 3 places first one with 1 and last two with 0 , so we have 2⁴ = 32 ways . So we finally get 192-32 ways = 160 ways .

So P(E) = Total no. Of outcomes for event / total no. Of outcomes (Sample Space)

So , P(E) =

=