What is the probability that this program will generate a byte where the first 2 bits are 1’s given that the byte contains an equal number of 0’s and 1’s?

Question

A byte is an 8-digit binary string where each digit, called a bit, is either a 0 of 1, e.g. 11010101. Suppose a computer program is written to randomly generate a byte where each bit is randomly assigned the value 0 or 1. What is the probability that this program will generate a byte where the first 2 bits are 1’s given that the byte contains an equal number of 0’s and 1’s? Enter your answer in the box below. HINT: To compute the intersection of the two events, calculate how many 1’s should occur in the last 6 bits when you know both that first two bits are 1’s and that the byte has an equal number of 0’s and 1’s.

Answer 1

11010101 where each bit has a probability of 1/2

The first 2 bits would be (1/2)(1/2) = 1/4

Each of the remaining bits have an equal probability of 1/2

Then (1/4)(1/2)⁶ = (1/4)(1/64) = 1/256