2.07A - Chapter 2, Section 7A Conditional Probability Game:...

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1 1 Chapter 2, Section 7A Conditional Probability © John J Currano, 01/12/2010 2 Game: Toss a die and observe the top face Sample Space: S = {1, 2, 3, 4, 5, 6} You win if the event A = {1, 2, 3} occurs. Now suppose that the die has been tossed and while you are not told what face shows, you are told the number is odd; i.e., that the event B = {1, 3, 5} has occurred. Would you still say P (win) = 0.5? Equivalently, would both players still be willing to bet even money? = 2 1 6 3 = = S A ) ( win) ( = A P P 3 Recall that in the unconditional case (no additional knowledge): Sample Space: S = {1, 2, 3, 4, 5, 6} You win if the event A = {1, 2, 3} occurs. 2 1 6 3 ) ( win) ( = = = = S A A P P Game: Toss a die and observe the top face Revised Sample Space: B = {1, 3, 5} You win if the event A B = {1, 3} occurs. 3 2 = B B A ) ( odd) | win ( = = B A P P 4 The conditional probability formula that we have derived works only for finite sample spaces in which all outcomes are equally likely (a uniform discrete probability model ). B B A B A P = ) ( Dividing both numerator and denominator of the fraction by | S | does not change its value and gives us a formula which we use for the definition of conditional probability : ) ( ) ( ) ( B P B A P S B S B A B B A B A P = = =
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2 5 ) ( ) ( ) | ( B P B A P B A P = Definition: The conditional probability
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2.07A - Chapter 2, Section 7A Conditional Probability Game:...

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