Step by step
The question
Question: the probability of bus is 0.7.
If bus, the probability of early is 0.6.
If no bus, it is 0.3.
Find the probability of early.
1 Draw the tree
The branches from one point add up to 1.
So no bus is 1 minus 0.7: 0.3.
And not early is 1 minus each: 0.4 and 0.7.
2 Multiply along the branches
Bus then early: we multiply along the branches.
0.7 times 0.6 is 0.42.
The same for every end.
No bus then early: 0.3 times 0.3 is 0.09.
3 Add the paths
Early can happen two ways: after bus, or after no bus.
We add the two paths.
Careful!
Multiply along each branch first.
0.6 only counts when it is bus.
0.42 plus 0.09 is 0.51.
4 Check
Let's check: the four ends add up to 0.42 plus 0.28 plus 0.09 plus 0.21.
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.