LESSON 6: PROBABILITY TREES

LESSON 6: PROBABILITY TREES Outline • Multiplication rule for two events • Multiplication rule for several events • Probability tree diagram and joint probabilities

MULTIPLICATION RULEFOR ANY TWO EVENTS • In the conditional probability rule the numerator is the probability that both A and B occur. It must be known in order to determine P(A|B). • However, in some applications P(A|B) and P(B) are known; in these cases we can multiply both side of the conditional probability formula by P(B) to obtain the multiplication rule. P(A and B) = P(A|B)P(B) • The conditional probability formula and the multiplication rule are both valid; in fact, they are equivalent.

MULTIPLICATION RULEFOR SEVERAL EVENTS • The multiplication rule for two components can be extended to several components. Consider events A1, A2, A3, …, An P(A1 and A2 and A3 … and An) = P(A1) × P(A2 | A1) ×P(A3 | A1 and A2) × … P(An|A1 and A2 and A3 … and An-1)

PROBABILITY TREES • Probability trees are useful to • calculate probabilities • identify all simple events • visualize the relationship among the events • Probability trees are useful if it is possible to • break down simple events into stages • identify mutually exclusive and exhaustive events at each stage • ascertain the probabilities of events at each stage

PROBABILITY TREES • A probability tree consists of some nodes and branches • Nodes • an initial unlabelled node called origin • other nodes, each labeled with the event represented by the node

PROBABILITY TREES • Branches • each branch connect a pair of nodes. • a branch from A to B implies that event B may occur after event A • each branch from • origin to A is labeled with probability P(A) • A to B is labeled with the probability P(B|A)

PROBABILITY TREES • Any path through the tree from the origin to a terminal node corresponds to one possible simple event. • All simple events and their probabilities are shown next to the terminal nodes.

PROBABILITY TREES • Example 1a: Construct a probability tree diagram for the Bendrix Company.

PROBABILITY TREES • Example 1b: Construct the joint probability table for the Bendrix Company.

PROBABILITY TREES • Example 2: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a probability tree diagram for this information.

PROBABILITY TREES • What is the probability of getting a red chip first and then a blue chip? • What is the probability of getting a blue chip first and then a red chip? • What is the probability of getting a red and a blue chip? • What is the probability of getting 2 red chips?

PROBABILITY TREES • Example 2 assumes that after an item (chip) is randomly selected in stage 1, the item is not returned (to the bag) before the next stage. This is a common assumption in sampling and is known as sampling “without replacement.” In this case an item cannot be selected more than once. • The assumption that simplifies computation is sampling “with replacement.” In this case, every item is returned immediately after it’s selected. So, the same item can be drawn several times. Example 3 uses this assumption. • Compare the answers of Examples 2 and 3. Are they very different?

PROBABILITY TREES • Example 3: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other with replacement. Construct a probability tree diagram for this information.

READING AND EXERCISES Lesson 6 Reading: Section 6-4, pp. 172-180 Exercises: 6-29, 6-30, 6-31

LESSON 6: PROBABILITY TREES