union… the union of two events is denoted if the event that occurs when either or both event...

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Union… Union… • The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B • We can use this concept to answer questions like: Determine the probability that a fund outperforms the market or the manager graduated from a top-20 MBA program.

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Page 1: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Union…Union…

• The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as:

A or B

• We can use this concept to answer questions like:

Determine the probability that a fund outperforms the market or the manager graduated from a top-20 MBA program.

Page 2: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Example Union…Determine the probability that a fund outperforms (B1)

or the manager graduated from a top-20 MBA program (A1).

B1 B2 P(Ai)

A1.11 .29 .40

A2.06 .54 .60

P(Bj) .17 .83 1.00

A1 and B1 occurs, A1 and B2 occurs, or A2 and B1 occurs…

P(A1 or B1) = .11 + .06 + .29 = .46

Page 3: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Example Union…Determine the probability that a fund outperforms (B1)

or the manager graduated from a top-20 MBA program (A1).

B1 B2 P(Ai)

A1.11 .29 .40

A2.06 .54 .60

P(Bj) .17 .83 1.00

P(A1 or B1) = .11 + .06 + .29 = .46

B1

A1

Page 4: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Probability Rules and Trees…

• We introduce three rules that enable us to calculate the probability of more complex events from the probability of simpler events…

1.The Complement Rule

2.The Multiplication Rule

3.The Addition Rule

Page 5: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Complement Rule…

• The complement of an event A is the event that occurs when A does not occur.

• The complement rule gives us the probability of an event NOT occurring. That is:

P(AC) = 1 – P(A)• For example:

in the simple roll of a die, the probability of the number “1” being rolled is 1/6.

The probability that some number other than “1” will be rolled is 1 – 1/6 = 5/6.

Page 6: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Multiplication Rule…Multiplication Rule…

• The multiplication rule is used to calculate the joint probability of two events. It is based on the formula for conditional probability defined earlier:

If we multiply both sides of the equation by P(B) we have:P(A and B) = P(A | B)•P(B)Likewise, P(A and B) = P(B | A) • P(A)If A and B are independent events, then P(A and B) = P(A)•P(B)

Page 7: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Addition Rule…Addition Rule…

• Recall: the addition rule was introduced earlier to provide a way to compute the probability of event A or B or both A and B occurring; i.e. the union of A and B.

P(A or B) = P(A) + P(B) – P(A and B)

Page 8: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Addition Rule…

• P(A1) = .11 + .29 = .40

• P(B1) = .11 + .06 = .17

• By adding P(A) plus P(B) we add P(A and B) twice. To correct we subtract P(A and B) from P(A) + P(B)

B1 B2 P(Ai)

A1.11 .29 .40

A2.06 .54 .60

P(Bj) .17 .83 1.00

P(A1 or B1) = P(A) + P(B) –P(A and B) = .40 + .17 - .11 = .46

B1

A1

Page 9: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

P(A1 or B2)

Find the probablity of :

P(A2 or B1) P(A2 or B2)

Page 10: Union… The union of two events is denoted if the event that occurs when either or both event occurs. It is denoted as: A or B We can use this concept to

Addition Rule for Mutually Excusive EventsAddition Rule for Mutually Excusive Events

• If and A and B are mutually exclusive the occurrence of one event makes the other one impossible. This means that

P(A and B) = 0• The addition rule for mutually exclusive events is

P(A or B) = P(A) + P(B) • We often use this form when we add some joint

probabilities calculated from a probability tree

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