7.5 Conditional Probability and Independence

This tutorial: Part B: Trees and Conditional Probability

Setting up a tree

Let us go back to the scenario discussed at the end of Part A (with different data):

You have invested in Home-Clone Inc. stocks, as the FDA is close to a decision as to whether to approve the company's "Clone-a-Sibling" kit. There is a chance of FDA approval, a chance that the stock will double if FDA approval is given, and a chance that the stock will double if FDA approval is not given.

These events seem to describe a two-stage "decision" process:

1. Will the FDA approve the kit? (Event A )
2. Will the stock double? (Event D )

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

Stage 1		Stage 2		Outcome
Start		A		D FDA approves and stock doubles
				D' FDA approves and stock does not double
		A'		D FDA does not approve and stock doubles
				D' FDA does not approve and stock does not double

Q What is the probability of each outcome?
A Since there is chance that the FDA will approve the kit,

P(A) = and so P(A') = .

Similarly, we are told that there is a chance that the stock will double if FDA approval is given. Thus:

P(D\|A) = which implies that P(D'\|A) =

Finally, we are told that there is a chance that the stock will double if FDA approval is not given. This gives:

P(D\|A') = which implies that P(D'\|A') = .

These probabilities can be added to the tree:

Start A D     D'     A' D     D'

Start A D   D'     A' D     D'

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

Finally, to obtain the probabilities of the outcomes (as listed on the right of the first tree diagram), we multiply all the probabilities along the path leading to each outcome.

Example The first outcome is: "FDA approves and stock price doubles," that is, D \cap A. Its probability is given by

P(D \cap A) = P(D\|A)\.P(A) =

Fill in the remaining probabilities and press "Check":

Start		A		D	P(D \cap A) =
				D'	P(D' \cap A) =
		A'		D	P(D \cap A') =
				D'	P(D' \cap A') =

   

Recall that D is the event that the stock will double. Referring to the above tree, what is the probability that the stock will double?

What is the probability that the stock will not double?

Notice that the probabilities we have just calculated give us the first stage of the tree we would get if we listed the two stages in reverse order:

Stage 1: Will the stock double? (Event D)
Stage 2: Will the FDA have approved the kit? (Event A)

Start D A     A'     D' A     A'

Start           D             A   A'     D'            A     A'

It now remains to calculate the four conditional probabilities on the right. These should be accurate to two decimal places: [Hint: Use the formula for conditional probability in each case; you already have the numerator and denominator in each case.]

	=		(Note: You can enter all answers as fractions.)
	=
	=
	=

We may not be given the information to create a tree directly, but may need to infer it from a indirect information like that given in a contingency table like the one in Part A of this tutorial that showed fictitious trial results of a new acne medication:

			Total
Total

Take:

M: Used the medication
I: Condition improved
N: No change
W: Condition got worse

Enter the probabilities in the following tree and press "Check".

Start M M' I N

W

I

N

W