$2.$ The tree in the figure below has probabilities after each chance node and PW values for each terminal node.

a$.$ What is the probability of $"$ H $"$ in uncertainty node $"$ C $1"?(5$ points$)$

b$.$ What is the expected value $($EV$)$ for each decision alternative "Pick X$"?(10$ points$)$

c$.$ What is the expected value for each decision alternative "Pick Y$"?(10$ points$)$

d$.$ What is the expected value for each decision alternative "Pick Z$"?(15$ points$)$

e$.$ What decision should be made and why? $(10$ points$)$