lean_workbook_plus_60316

Let $p$ be the probability that $\lceil\log_4x\rceil$ is odd when $x$ is randomly chosen from the interval $(0,1]$ . We can see that this is the same as the probability that $\lceil\log_4(4x)\rceil$ is odd when $x$ is randomly chosen from the interval $\left(0,\frac14\right]$ . Furthermore, $\lceil\log_4(4x)\rceil=\lceil\log_4x+1\rceil=\lceil\log_4x\rceil+1$ , which shows that $\lceil\log_4(4x)\rceil$ is odd if and only if $x$ is even. Hence, the probability that $\lceil\log_4x\rceil$ is odd when $x$ is randomly chosen from the interval $\left(0,\frac14\right]$ is equal to $1-p$ . Therefore, the probability that $\lceil\log_4x\rceil$ is odd when $x$ is randomly chosen from the interval $(0,1]$ can also be expressed as: $p=\frac14(1-p)$ Solving for $p$ gives us $p=\frac15$ . Therefore, the probability that $y$ is odd when $x$ is randomly chosen from $(0,1]$ is $\boxed{\frac15}$ .