Conditional Probability, Lecture notes of Probability and Statistics

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Conditional Probability

By: Engr. Marc Rivera

The probability of an event B occurring

when it is known that some event A has

occurred is called a conditional probability

and is denoted by P(B|A).

Conditional Probability

The conditional probability of B, given A,

denoted by P(B|A), is defined by

provided P(A) > 0

Conditional Probability

Events: M – a man is chosen

E – the one chosen is employed

Find P( M | E ).

Illustration Employed Unemployed Total Male 460 40 500 Female 140 260 400 Total 600 300 900

Events: M – a man is chosen

E – the one chosen is employed

Find P( E | M ).

Illustration Employed Unemployed Total Male 460 40 500 Female 140 260 400 Total 600 300 900

The probability that a regularly scheduled flight departs on time is P(D)=0.83; the probability that it arrives on time is P(A)=0.82; and the probability that it departs and arrives on time is P(D∩A) = 0.78. Find the probability that a plane arrives on time, given that it departed on time, and (b) departed on time, given that it has arrived on time. Example

From previous example, the probability that it arrives on time, given that it did not depart on time Example

1. 24

1. 08

Two events A and B are independent if and only if: P(B|A) = P(B) or P(A|B) = P(A) Independent Events

If in an experiment the events A and B can both occur, then P(A∩B) = P(A) P(B|A), provided P(A) > 0. The Product Rule, or the Multiplicative Rule

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One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black? Example