Multiplication%20and%20Addition%20Rules%20for%20Probability

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Multiplication and Addition Rules for Probability

• Special Topics

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General Addition Rule

• Last time, we learned the Addition Rule for
  Mutually Exclusive events (Disjoint Events). This
  was
• P(A or B) P(A) P(B). This is used when events
  A and B cant happen at the same time, such as
  getting a head or a tail on a single coin flip.
  You cant get both a head and a tail on the same
  coin flip!
• There are other events which can happen at the
  same time, such as being a senior and being a
  student in Special Topics.

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General Addition Rule

• Lets let event A be the event that a person is a
  senior. Lets call P(A) .25
• Lets let event B be the event that a person is
  in Special Topics. Lets call P(B) .8
• Lets also call P(A and B) .10
• If we go by the Addition Rule for Disjoint
  Events, then P(A 0r B) .25.8 1.05 (what is
  wrong with this?).
• The problem here is that a person can be both a
  senior and a Special Topics student at the same
  time. When this happens, things get counted twice!

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A Diagram to Sort This Out!

• This is the situation in a Venn Diagram
• Notice the intersection gets counted twice! We
  need a formula that corrects for this.

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General Addition Rule

• Here is the formula for the General Addition
  Rule
• P(A or B) P(A) P(B) P(A and B). We subtract
  one version of the joint probability since it
  gets counted twice.
• When event are disjoint, the P(A and B) part of
  the formula equals 0. That takes us back to the
  formula for disjoint events P(A or B) P(A)
  P(B).

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Example

• Musical styles other than rock and pop are
  becoming more popular. A survey of college
  students finds that the probability they like
  country music is .40. The probability that they
  liked jazz is .30 and that they liked both is
  .10. What is the probability that they like
  country or jazz? (Hint a joint probability is
  given, so the General Addition Rule Applies!)
• P(C or J) .4 .3 -.1 .6

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The Multiplication Rule

• What do we do to calculate the probability of two
  (or more) independent events happening at the
  same time?
• We use the Multiplication Rule for Independent
  Events. The key word is the word and.
• The formula is P(A and B) P(A)P(B)
• What is the probability of flipping a coin and
  getting a head, and rolling a die and getting a
  six?

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Independence

• Definition Two events are independent when
  knowing that one occurred does not change the
  probability that the other occurred.
• Coin flips and dice rolls are independent.

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Homework

• Section 8.1 Part 2 worksheet.