Chapter 15: Probability Rules

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Chapter 15 Probability Rules!
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Vocabulary

• Event- any set or collection of outcomes
• Sample space- collection of all possible
  outcomes denoted S
• Disjoint events- two events have no
  outcomes in common
• Independent- two events are independent
  if knowing whether one event occurs
  does not alter the probability that the
  other event occurs
• Tree Diagram- A display of conditional events or
  probabilities helpful in thinking
  through condition

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Events

• When the k possible outcomes are equally likely,
  each has a probability of 1/k.
• For any even A, that is made up of equally likely
  outcomes, P(A)count of outcomes in A
• count of all possible outcomes
• Remember, this formula is only possible when all
  the outcomes are equally likely

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The First Three Rules for Working with
Probability Rules

• Make a picture, Make a picture, Make a picture
• The most common type of picture to use with
  probability would be a Venn diagram

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The General Addition Rule

• In this rule
• We add the probabilities of the two events
• Then subtract out the probability of their
  intersection
• P(A U B)P(A) P(B) P(A n B)

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It Depends

• When we want the probability of an event from a
  conditional distribution we write
• P(B l A) pronounced- B given A
• A probability that takes into account a given
  condition such as this is called a conditional
  probability
• To obtain the probability of the even B given the
  even A, we focus in on the outcomes in A
• We then find in what fraction of those outcomes
  also occurred written
• P(B l A) P(A n B)
• P(A)
• The formula only works when P(A) gt 0

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The General Multiplication Rule

• When A and B independent rule
• P(A n B) P(A)P(B P(B)
• The general multiplication
• P(A n B) P(A) x P(B l A)
• Events dont have to be independent

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Independence/ Independent ? Disjoint

• Events A and B are independent when
• P(B l A) P(B)
• Disjoint events are not independent
• Lets use grades as an example, if you get an A in
  a course, then the probability of getting a B is
  now 0
• Knowing you got an A changed the probability of
  getting a B to 0
• Mutually exclusive events cant be independent

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Depending on Independence

• Be careful in assuming the independence of events
• Whenever you see yourself multiplying it over and
  over, make sure to stop and think whether the
  events are truly independent

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Drawing Without Replacement

• Lets say your trying to pick a red card out of a
  standard deck of playing cards
• P(R) 26/52
• Now once youve drawn throw the card aside and
  try picking a red card again
• P(R) 25/51
• You have to change youre total and possible
  outcomes to fit the circumstances
• P (R n R) 26/52 x 25/51 .245

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Tree Diagrams

• Very helpful tool in probability is drawing a
  picture as shown before in Venn Diagrams
• Now we have tree diagrams which are also very
  helpful
• From Left to right top to bottom or bottom to top
  a table is made stating all possible outcomes
  from a certain point
• Tree diagrams can be done in several ways and
  with several different things, here is one way

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Tree Diagrams
.42
.21
.5
.29
.58
.33
.165
.5
.335
.67
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Reversing the Conditioning

• If we were to use the Tree Diagram from before
  and ask for P(A l B) it would not be the same
• First you must take P(A n B) and divide by P(B)
  or P(A l B) P(A n B) / P(B)
• So .5 x .42 .21
• Then divide by .21 .165 .375
• And the P(A l B) .21/.375 .56

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Bayers Rule

• When we have P(A l B) but want the reverse
  probability P(B l A) wee need to know
• P(A n B)
• P(A)
• A tree can help us easily find these values
• The actual rule is
• P(B l A) P(A l B)P(B) _
• P(A l B)P(B) P(A l Bc)P(Bc)

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What Can go Wrong?

• Dont use a simple probability rule where a
  general rule is appropriate
• Dont assume independence
• Dont assume disjoint without proving
• Dont find probabilities for samples drawn
  without replacement as if they had been drawn
  with replacement
• Adjust the denominator
• This adjustment when small population or a large
  portion of a finite population

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What Can go Wrong?

• Dont reverse conditioning naively
• The probability of B given A may not, and in
  general, does not, resemble the probability of A
  given B
• Dont confuse disjoint with Independent
• Disjoint cant happen simultaneously, when one
  occurs the other cant P(B l A)0
• Independent events must be able to occur at the
  same time, when one happens you know it has no
  effect on the other, so P(B l A)P(B)