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Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ and $Z=\left\{\begin{array}{ll}1 & \text { if } X_{1}=Y \\ 2 & \text { otherwise }\end{array}\right.$.

Which of the following is true?

  1. $\operatorname{Pr}[Z=1]=\operatorname{Pr}[Z=1 \mid Y \geq 0.3] $
  2. $\operatorname{Pr}[Z=1]=\operatorname{Pr}\left[Z=1 \mid Y=0.3, X_{1}=0.2\right]$
  3. $\operatorname{Pr}[Z=1]=\operatorname{Pr}\left[Z=1 \mid Y=0.3, X_{1}=0.3, X_{2}=0.2\right]$
  4. $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$
  5. $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$
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