Probabilistic and Statistical Techniques 1 Lecture 9 Dr. Nader Okasha

2 Mathematics of Probabilities

3 Axioms of Probability

4 Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  For any event A, the probability of A is between 0 and 1 inclusive. That is, 0  P(A)  1.

5 Addition Rule

6

7 Disjoint events Events A and B are disjoint if they cannot occur at the same time. (That is, disjoint events do not overlap.) Events That Are Not DisjointDisjoint Events

8 Rules of Complementary Events

If P(A) = 0.3 P(A) = 1 – P(A) = 1 – 0.3 = Venn Diagram for the Complement of Event A

10 Example 1 When two balanced dice are rolled, 36 equally likely outcomes are possible: a) find The probability the sum is 11, b) the two dice are doubles The sum of the dice can be 11 in two ways. The probability the sum is 11 = 2/36 = Doubles can be rolled in six ways. The probability of doubles = 6/36 =

11 Example 2 Define:

12 Example 2 AABACA BABBCB CABCCC E1E1 E1E1

13 Example 3 Define:

14 Example 3

Lecture 815 Example 4 Titanic Passengers (Table 3-1), Assuming that 1 person is randomly selected from 2223 people abroad the titanic: Find P (selected a man or a boy) Find P (selected a man or some one who survived)

16 Solution P (selected a man or a boy) = P (men) + P(boys) P (selected a man or survived) = P (men) + P (survived) – P (men & survived)