Combination of exponentional and uniform distribuion problem

Question

Let $X$ be uniform distributed in $[0, 1]$. Let $Υ$ be exponenential distributed with $θ = 1$. Let $Y$ that is defined by tossing a fair coin and if tails comes up then $Z=X$, else if heads comes up, then $Z=Y$. Any hint on how can i calculate $P(Z<0.5)$?

Answer 1

Hint: $$ \mathbb P(Z < 1/2) = \mathbb P(Z < 1/2 \mid \text{heads})\times\mathbb P(\text{heads}) + \mathbb P(Z < 1/2 \mid \text{tails})\times\mathbb P(\text{tails}).$$