P(test) = 0.5, P(pass given test) = 0.9, P(pass given no test) =…

Step by step

The question

Question: the probability of test is 0.5.

If test, the probability of pass is 0.9.

If no test, it is 0.6.

Find the probability of pass.

1 Draw the tree

The branches from one point add up to 1.

So no test is 1 minus 0.5: 0.5.

And not pass is 1 minus each: 0.1 and 0.4.

2 Multiply along the branches

Test then pass: we multiply along the branches.

0.5 times 0.9 is 0.45.

The same for every end.

No test then pass: 0.5 times 0.6 is 0.3.

3 Add the paths

Pass can happen two ways: after test, or after no test.

We add the two paths.

Careful!

Multiply along each branch first.

0.9 only counts when it is test.

0.45 plus 0.3 is 0.75.

4 Check

Let's check: the four ends add up to 0.45 plus 0.05 plus 0.3 plus 0.2.

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.