## 7.6 Conditional Probability and Independence

### Part B: Trees and Conditional Probability

(Based on Section 7.6 in Finite Mathematics and Finite Mathematics and Applied Calculus)

Note There should be navigation links on the left. If you got here directly from the outside world and see nothing on the left, press here to bring up the frames that will allow you to properly navigate this tutorial and site.

(For best viewing, adjust window width to match the length of the line below.)

### Setting up a tree

Let us go back to the stock price question in Part A, and add some additional information.

You have invested in Home-Clone Inc. stocks, as you suspect that the company's "Clone-a-Sibling" kit will shortly be approved by the FDA. There is an 80% chance that FDA approval will be given, and a 95% chance that the value of the stock you hold will double if FDA approval is given. If FDA approval is not given, then there is only a 10% chance of your stock doubling.

This seems to describe a two-stage "decision" process:

1. Will the FDA approve the kit?
2. Will the stock double?

We can model this using a "decision" tree which branches twice; once for each decision stage shown above.

 Stage 1 Stage 2 Outcome Start FDA approves & stock price doubles FDA approves & stock price does not double FDA does not approve & stock price doubles FDA does not approve & stock price does not double

 Key: FDA approves FDA does not approve Stock price doubles Stock price does not double

Q What are the probabilities of each outcome?
A Since there is 80% chance that the FAD will approve the kit, the probability of approval is 0.8, and the probability of disapproval is therefore 0.2. These probabilities can be added to Stage 1 of the tree:

 Stage 1 Stage 2 Outcome Start FDA approves & stock price doubles FDA approves & stock price does not double FDA does not approve & stock price doubles FDA does not approve & stock price does not double

Note that the sum of the probabilities leaving the start node is 1.

Now for Stage 2. We are told that, if FDA approval is given (putting us at the "A" node), the probability of the stock doubling is 0.95 (95%). This allows us to fill in the branches on the upper right.

On the other hand, if we are at the "NA" node (FDA dfoes not approve) then the probability of the stock doubling is only 0.1 (10%). This allows us to fill in the remaining branches:

 Stage 1 Stage 2 Outcome Start FDA approves & stock price doubles FDA approves & stock price does not double FDA does not approve & stock price doubles FDA does not approve & stock price does not double

Again, notice that the sum of the probabilities on the arrows leaving each node is 1.

Finally, to obtain the probabilities a specific outcome, we mutliply all the probabilities along the path leading to that outcome

Example The first outcome is: "FDA approves & stock price doubles." It probability is given by (0.8)(0.95) = 0.76.

Here is the completed tree diagram:

 Stage 1 Stage 2 Outcome Probability Start FDA approves & stock price doubles 0.76 FDA approves & stock price does not double 0.04 FDA does not approve & stock price doubles 0.02 FDA does not approve & stock price does not double 0.18 Sum: 1.0

 Key: FDA approves FDA does not approve Stock price doubles Stock price does not double

Referring to the above tree, P(ND|A) = ?

 0.05 0.04 0.2 0.95

Which of the following conditional probabilities already appear on the tree?

 P(A|D) P(D|NA) P(D|A) P(A|ND)

Here once again is the tree (to save the bother of scrolling back and forth).

 Stage 1 Stage 2 Outcome Probability Start FDA approves & stock price doubles 0.76 FDA approves & stock price does not double 0.04 FDA does not approve & stock price doubles 0.02 FDA does not approve & stock price does not double 0.18 Sum: 1.0

Referring to the above tree, the probability that the stock did not double, is?

 0.0072 0.95 0.22 0.045

Referring to the above tree, the probability that the FDA did not approve the kit, given that the stock did not double, is?

 1/9 11/50 9/10 9/11

Here is the table from Part A showing a fictitious statistical study of a new acne cream.

 Used Cream Used Placebo Skin Improved 800 600 No Improvement 400 200

Which of the following trees correctly models the scenario?

 B B
 B B
 B B
 B B

You now have several options:

• Go on to Part C by pressing the "Next Tutorial" button on the left
• Try some of the questions in the true/false quiz (warning: it covers the whole of chapter 76) by pressing "Chapter Quiz" on the left
• Try some of the exercises in Section 7.6 of Finite Mathematics, or Finite Mathematics and Applied Calculus.

Last Updated: July, 2000