Question 1161051
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You have the correct answer. Nice work.


Here is the original tree diagram. I redrew it to just focus on vertices (aka points) and edges (aka connecting lines). The other parts of the drawing such as the grass are just filler/distraction in my opinion.
<img width="35%" src = "https://i.imgur.com/htqm7AW.png">


For each decision branch or junction point, label the probabilities for the edges. The first junction has 3 choices, so assuming each path is equally likely we have probability 1/3 for each path. A similar situation happens with the other junctions.
<img width="35%" src = "https://i.imgur.com/Dl6zvIS.png">


Starting from each "cash" block on the right side, highlight out a path back to the starting point. This guarantees we win cash. 
<img width="35%" src = "https://i.imgur.com/jpH7Unz.png">
Whatever values are highlighted along the way are multiplied out. We can multiply the probabilities because the events of selecting any given path is independent from any other path selection.


Red path = (1/3)*(1/2)*(1/2) = 1/12
Red path,then blue path = (1/3)*(1/2) = 1/6
green path, then blue path = (1/3)*(1/2) = 1/6
purple path = (1/3)*(1/2)*(1/3) = 1/18
All of these paths describe a way to get a cash prize. Every other path leads to no cash prize.


Adding up those probabilities gives
1/12+1/6+1/6+1/18 = 3/36+6/36+6/36+2/36 = (3+6+6+2)/36 = 17/36


So the probability of winning a cash prize is 17/36 = 0.47222222222222
This rounds to 0.472
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