Probabilistic and Statistical Techniques

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Lecture 9

Dr. Nader Okasha

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Mathematics of Probabilities

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Axioms of Probability

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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.

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Addition Rule

B)P(AP(B)P(A)B)P(A

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Addition Rule

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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 Disjoint

Disjoint Events

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Rules of Complementary Events

)(1)(

)(1)(

1)()(

APAP

APAP

APAP

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If P(A) = 0.3

P(A) = 1 – P(A) = 1 – 0.3 = 0.7

Venn Diagram for the Complement of Event A

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Example 1When 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 = 0.056.

Doubles can be rolled in six ways. The probability of doubles = 6/36 = 0.167.

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Example 2

Define:

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Example 2AA BA CA

BA BB CB

CA BC CC

E1E1

B)P(AP(B)P(A)B)P(A

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Example 3

Define:

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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)

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Solution

P (selected a man or a boy) = P (men) + P(boys)

1692 640.79

2223 2223

P (selected a man or survived) = P (men) + P (survived) – P (men & survived)

1692 706 3320.929

2223 2223 2223