Step by step
The question
Question: the probability of rain is 0.3.
If rain, the probability of late is 0.4.
If no rain, it is 0.1.
Find the probability of rain, given late.
1 Draw the tree
The branches from one point add up to 1.
So no rain is 1 minus 0.3: 0.7.
And not late is 1 minus each: 0.6 and 0.9.
2 Multiply along the branches
Rain then late: we multiply along the branches.
0.3 times 0.4 is 0.12.
The same for every end.
No rain then late: 0.7 times 0.1 is 0.07.
3 Add the paths
Late can happen two ways: after rain, or after no rain.
We add the two paths.
Careful!
Multiply along each branch first.
0.4 only counts when it is rain.
0.12 plus 0.07 is 0.19.
4 Divide by the given
Given late: only the two late paths count, 0.19.
Our path, rain then late, is 0.12.
0.12 divided by 0.19 is about 0.632, to 3 decimal places.
5 Check
Let's check: the four ends add up to 0.12 plus 0.18 plus 0.07 plus 0.63.
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.
Then: divide by the given.
Last: check.