Trees

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Trees
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Oak Trees
Pine Trees
Probability Trees
Decision Trees
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Compound Events
Two or more events happening at the same time.
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Conditional Probability
If the probability of Event B occurring is
dependent on whether Event A has occurred, you
would say that Event B is conditional on Event A
Conditional Probability
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Conditional Probability

• If the probability of Event B occurring is
  dependent on whether Event A has occurred, you
  would say that Event B is conditional on Event A,
  and is written
• P(BA)
• which means the probability of B given that A
  has occurred

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Multiplication Rule

• Multiplication rule for dependent events
• P(A and B) P(A) X P(BA)
• If two (or more) events are independent then
• P(A and B) P(A) X P(B)

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Probability Trees
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Tree diagrams

• A very useful diagram to use when solving
  compound events, particularly when conditional
  probability is involved, is the tree diagram.
• This diagram represents different outcomes of an
  experiment by means of branches.
• The event and probability are written alongside
  the branch

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Probability tree

• The two experiments of choosing an individual can
  be represented by the tree diagram
• The first experiment is represented by a small
  circle or node, and the two possible outcomes by
  branches.

Chance branches
Chance Nodes
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Student Tree Diagram
The routes are mutually exclusive and mutually
exhaustive
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Student Tree Diagram
Two girls 0.2778
girl
2nd
0.5
boy
girl
Girl and boy 0.2778 0.2778
1st
0.556
0.5
boy
girl
2nd
0.625
0.444
boy
Two boys 0.1667
0.375
The routes are mutually exclusive and mutually
exhaustive
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Question 3

• A company purchases electronic components in
  batches of 100 and the supplier guarantees that
  there will be no more than 5 defective component
  in each batch.
• Before acceptance of a particular batch the
  company has a policy of selecting without
  replacement two components for testing.

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Question 3 (Continued)

• If both components are satisfactory the batch is
  accepted and if both are defective the batch is
  rejected.
• However, if only one is defective another
  component is selected and if it is defective, the
  batch is rejected. If the probability that a
  component is defective is 5, what is the
  probability that the batch will be accepted?

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Q 3 Solution
0.90202 Accept
OK
94/99
0.04602 Accept
OK
5/99
OK
94/98
Defective
4/98
95/100
0.00916 Reject
Defective
0.04602 Accept
OK
5/100
94/98
OK
4/98
Defective
0.00196 Reject
Defective
95/99
4/99
0.00202 Reject
Defective
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Q 3 - Solution Notes

• There are three routes where the decision is to
  accept, and the addition law can be used to give
  the probability that the batch will be accepted.
  That is
• 0.90202 0.04602 0.0462 0.99406
• The probability that the batch would be rejected
  without cause is very small
• 1 - 0.99406 0.00594

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Expected Value
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What is Expected Value?

• It is often possible to assign probabilities to
  future states of nature. This is decision making
  under conditions of risk.
• The most widely used decision making criterion
  under risk is expected value.In general
• Expected value Spx
• where xoutcome pprobability of
  outcome
• where S means the sum of

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Activity 1

• Over a long period of time a car salesperson
  recorded the number of sales she achieved per
  day. From an analysis of her records it was
  found that she made
• no sales 10 of the time
• one sale 10 of the time
• 2 sales 50 of the time
• 3 sales 30 of the time
• What is her expected number of sales

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Activity 1 Solution

• The x in this case takes on values of 0,1, 2,3.
  The expected value is
• 0.1 x 0 0.10x1 0.50x20.30X3
• 0 0.10 1.00.90
• 2.0 sales
• In general the expected value is the long-run
  average

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Decision Trees
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What is a decision tree?

• Decision trees are similar to probability trees
  except that as well as probabilistic (or chance)
  branches they also have decision branches.
• Decision branches allow the decision-maker to
  compare alternative options, while the chance
  branches handle the probabilistic nature of an
  outcome.

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Decision Tree Skeleton
Chance Nodes
Decision Node
Chance branches
Decision branches
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Using Decision Tree

• The decision tree is drawn from left to right,
  but to evaluate the tree you work from right to
  left.
• This is called the roll-back method
• You first evaluate the EMV at each chance node,
  and then at the decision node you select the best
  EMV.

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Example

• Dubai Car accessory company has a distribution
  decision to make. They can market the product
  nationwide, sell by mail order or sell the patent
  for their invention.
• The estimated profits for each decision depend on
  the state of the market, which will be either
  high (0.25), medium (.03) or low (0.45).

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Example Decision Tree
Expected profits AED 000 for Dubai Car Company
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Example Decision Tree

• What decision should the company make in order to
  maximize their EMV?
• Note that this is a single-stage decision
  problem. Decision trees are normally used where
  multi-stage decisions are involved.

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Summary

• Probability can be used to calculate expected
  monetary values, which are used in solving
  decision problems
• Decision trees are one method of solving decision
  problems, and these are commonly used when one
  decision leads to another.