ITM 107 Lecture Notes - Lecture 13: Sample Space, Conditional Probability, Product Rule
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LECTURE 10 INTRODUCTION TO PROBABILITY
• Each probability that we have computed has been relative to the sample space for the
• Sometimes information is given that reduces the sample space needed to solve a stated
• For example, if a die is rolled, the probability that a 5 occurs, given that an odd number
is rolled, is denoted by
Pr(5 rolled | odd number rolled)
• The knowledge that an odd number occurs reduces the sample space from the numbers
on a die, , to the sample space for odd numbers on a die,
, so the event E = “a 5 occurs” has probability
• To find Pr(A | B), we seek the probability that A occurs, given that B occurs.
• Figure 7.4 shows the original sample space S with the reduced sample space B shaded.
• Because all elements must be contained in B, we evaluate Pr(A | B) by dividing the
number of elements in A B by the number of elements in B.
• Condition Probability
• Using the definition of Pr(A | B) and dividing the numerator and denominator by n(S),
the number of elements in the sample space, gives