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09-28 Conditional Probability and Independence (1)

09-28 Conditional Probability and Independence (1) - BTRY...

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BTRY 4080 / STSCI 4080 Fall 2010 128 Conditional Probability and Independence Chapter 3, Sections 3.1 - 3.5
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BTRY 4080 / STSCI 4080 Fall 2010 129 Example: Suppose we toss a fair dice twice . What is the probability that the sum of the 2 dice is 8 ? Sample space S = { ?? } Event E = { ?? } Suppose I tell you that the first dice landed on a 3 ; what is the probability that the sum of the 2 dice is 8 ? (Reduced) S = { ?? } (Reduced) E = { ?? }
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BTRY 4080 / STSCI 4080 Fall 2010 130 Basic Formula of Conditional Probability We have two events A, B S . We define: P ( A | B ) = P ( AB ) P ( B ) , if P ( B ) > 0 P ( A | B ) : conditional probability that A occurs, given that B occurs.
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BTRY 4080 / STSCI 4080 Fall 2010 131 Venn diagram: A and B both occur: light gray area B occurs: union of light blue and light gray areas Probability P ( A | B ) : determine frequency of AB , relative to frequency of B
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BTRY 4080 / STSCI 4080 Fall 2010 132 Example (revisited): Suppose we toss a fair dice twice. Calculation on Slide 129 uses a “reduced sample space” approach – we determine how the composition of the sample space and event of interest changes upon introducing the condition that the first die lands on a 3; we then calculate the desired probability . Direct approach: let A = “sum of dice is 8” and B = “first die lands on 3”. Then: P ( A | B ) = P ( AB ) P ( B ) = 1 36 6 36 = 1 6 . Note that order of appearance of events relative to | matters: P ( B | A ) = P ( AB ) P ( A ) = 1 36 5 36 = 1 5 .
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BTRY 4080 / STSCI 4080 Fall 2010 133 Example (3.2.2c): In the card game bridge, all 52 cards are dealt out equally to 4 players — called East, West, North, and South. If North and South have a total of
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