Step by step
The question
Question: the probability of win is 0.4.
If win, the probability of party is 0.9.
If no win, it is 0.2.
Find the probability of party.
1 Draw the tree
The branches from one point add up to 1.
So no win is 1 minus 0.4: 0.6.
And not party is 1 minus each: 0.1 and 0.8.
2 Multiply along the branches
Win then party: we multiply along the branches.
0.4 times 0.9 is 0.36.
The same for every end.
No win then party: 0.6 times 0.2 is 0.12.
3 Add the paths
Party can happen two ways: after win, or after no win.
We add the two paths.
Careful!
Multiply along each branch first.
0.9 only counts when it is win.
0.36 plus 0.12 is 0.48.
4 Check
Let's check: the four ends add up to 0.36 plus 0.04 plus 0.12 plus 0.48.
That is 1: every way is counted once.
So the answer is right!
Let's look back.
First: draw the tree.
Then: multiply along the branches.
Then: add the paths.
Last: check.