# ITM 107 Lecture Notes - Lecture 13: Sample Space, Conditional Probability, Product Rule

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10 Dec 2018

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KATIE CHOI

12/3/18

LECTURE 10 INTRODUCTION TO PROBABILITY

Conditional Probability

• Each probability that we have computed has been relative to the sample space for the

experiment.

• Sometimes information is given that reduces the sample space needed to solve a stated

problem.

• 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.

(Guergachi, 2018)

• 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

(Guergachi, 2018)

• 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