Conditional Probabilities

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Conditional Probabilities
Math 2510 Spring 2004
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Review the story so far

• We have a sample space, and for each x in the
• sample space, we have the probability that we
• observe x on any given experiment. This
• probability is assigned without having any
• additional information about the experiment.

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However, sometimes we have information about the
outcome. We do not know the exact outcome, but
maybe we have some inside information. Consider
the following example Suppose we are talking
about drawing cards from a deck. Let Aget a red
suit and let Bget a diamond. Suppose someone
pulls a card from a deck and asks what is P(B)?
We know it is 13/54, but assume that the person
asking the question was careless when they drew a
card, and you saw that it was in fact a red card.
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• If we know the card drawn was red, then it was
• either a heart or a diamond. In that case we can
  
• say the probability is ½!

So, knowing partial information about the outcome
alters the original probability! This is the key
idea behind conditional probability!
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Conditional Probability
We define the probability of the event B given
that the event A occurred as
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Lets use this notation to compute the previous
example Aget a red suit Bget a
diamond. P(BA)P(A n B)/P(A)P(B)/P(A)(1/4)/(1/
2)1/2
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Example 1
To develop programs for business travelers, a
company commissioned a study of executives that
play golf. The study revealed that 55 of the
execs admitted that they cheated when they play
golf. Also, 20 admitted they cheated at golf
and lied in business. Given that an executive
cheated at golf, what is the probability that the
executive had also lied in business?
Answer 0.2/0.550.364
We want
Bgolf and business Agolf
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Close

• The conditional probability of the event A given
  the event B is
• To use this, just identify A and B and plug them
  in!