Probability and Probability and Probability and Probability and Rules on ProbabilityRules on ProbabilityRules on ProbabilityRules on Probability

• Introduction • Experiments, Outcomes

and Sample Spaces • Operations with Events • Probability of an Event • Rules on Probability

Chapter Outline

Chapter

8

Probability

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Introduction In general, the word probability is used to define the chance of occurrence of an event. A lot of people are familiar with probability from observing games of chance, such as card games and lotteries. Probability is not only used in games of chances. Predictions in the fields of weather forecasting, investments, insurance, and other areas rely on probability. This chapter introduces the basic concepts of probability and the different rules used to solve problems involving probability. Experiments, Outcomes and Sample Spaces An experiment or statistical experiment, is the process that when performed, results in one and only one of many observations. These observations are called outcomes. The collection of all outcomes for an experiment is called a sample space, and is usually denoted by S . Example 1

Determine the sample space in each experiment. a. rolling a die once b. tossing a coin twice c. a coin is tossed until a tail or three heads appear

Solution

Example 2

A newly married couple is planning to have three children. List the elements of the sample space 1S using the letter M for “male” and the letter F for “female”. Define a second sample space 2S where the elements represent the number of females.

Solution

Probability

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Example 3

An experiment involves tossing a pair of dice, 1 green and 1 red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space.

Solution

An event is a subset of the sample space. If an event is a set containing only one element of the sample space, then it is called a simple event. A compound event is one that can be expressed as the union of simple events. Two events A and B are mutually exclusive if they cannot occur together. Two events are said to de independent if the occurrence of one does not affect the occurrence of the other. The null space or empty space is a subset of the sample space that contains no elements. We denote this event by the symbol ∅. Example 4

For the sample space of Example 3, a. List the elements corresponding to the event A that the sum is greater than 8 b. List the elements corresponding to the event B that a 2 occurs on either die. c. List the elements corresponding to the event C that a number greater than 4 comes up on the green die.

Solution

Probability

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Operations with Events 1. Intersection of Events

The intersection of two events A and B, denoted by the symbol BA∩ , is the event containing all elements that are common to A and B.

2. Union of Events

The union of two events A and B, denoted by the symbol BA∪ , is the event containing all the elements that belong to A or to B or to both.

3. Complement of an Event

The complement of an event A with respect to S is the set of all elements of S that are not in A. We denote the complement of A by the symbol 'A .

Example 5

For the sample space of Example 3, a. List the elements corresponding to the event CA∩ b. List the elements corresponding to the event BA∩ c. List the elements corresponding to the event 'C

Solution

Probability of an Event Probability refers to the likelihood of occurrence of an event. There are three approaches to probability. The subjective probability refers to the probability in which chance of occurrence given by a particular person to an event is based on his/her educated guesses, opinion, intuition or beliefs. The empirical probability is the one in which probability is assigned based on the prior knowledge of the events that happened in the past or based on research or experiment. The classical probability approach is applied only when all possible outcomes are equally likely to happen, and the probability of an event is equal to 1 divided by the total number of outcomes.

Probability

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Probability is a numerical measure of the likelihood that a specific event will occur. The probability of an event always lies in the range 0 to 1. To every point in the sample space we assign a probability such that the sum of all probabilities is 1. The probability that an event E will occur is denoted by )E(P and the probability that it will not occur is denoted by )'E(P . An event will either occur or not occur. Hence, the sum of the probability of occurrence and the probability of nonoccurrence is always equal to 1. That is, 1=+ )'E(P)E(P . The probability of an event that is certain to occur is 1. The probability of an event that cannot occur is zero.

If an experiment can result in N equally likely outcomes and an event E can

result in noutcomes then the probability that E will occur is Nn. More formally,

N)E(n

)E(P =

where =)E(n number of sample points in E =N total number of sample points in the sample space

Example 6

Find the errors in each of the following statements: a. The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given day in February are, respectively, 0.19, 0.38, 0.29, and 0.15.

b. The probability that it will rain tomorrow is 0.40 and the probability that it will not rain tomorrow is 0.52. c. The probabilities that a printer will make 0, 1, 2, 3, or 4 or more mistakes in printing a document are, respectively, 0.19, 0.34, -0.25, 0.43, and 0.29. d. On a single draw from a deck of playing cards the probability of selecting a heart is ¼, the probability of selecting a black card is ½, and the probability of selecting a black heart card is ¼.

Solution

Probability

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Example 7

a. Find the probability of getting an even number from a single roll of a dice. b. If a card is drawn from an ordinary deck, find the probability of drawing i. an ace ii. a spade iii. a face card c. A box contains 5 red, 4 blue, and 3 white balls. If a ball is chosen at random, what is the probability that i. it is not red ii. it is not white

Solution

Example 8

If a letter is chosen at random from the English alphabet, find the probability that the letter

a. is a vowel; b. precedes the letter j; c. follows the letter g.

Solution

Probability

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Example 9

A box contains 5 red, 4 blue, and 3 white balls. Suppose three balls are drawn at random. What is the probability that

a. they are of different colors? b. they are all red? c. two are blue, one is white? d. exactly two are blue? e. none is blue?

Solution

Example 10

a. A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is an event that a number less than 4 occurs on a single toss of the die, find )E(P .

b. Three men are seeking public office. Candidates A and B are given about the same chance of winning, but candidate C is given twice the chance of either A or B. i. What is the probability that C wins? ii. What is the probability that A does not win?

Solution

Probability

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Example 11

If a permutation of the word “white” is selected at random, find the probability that the permutation a. begins with a consonant; b. ends with a vowel; c. has the consonants and vowels alternating.

Solution

Example 12

If 3 books are picked at random from a shelf containing 5 novels, 3 book of poems, and a dictionary, what is the probability that

a. the dictionary is selected? b. 2 novels and 1 book of poems are selected?

Solution

Rules on Probability Theorem 1 (Addition Rule) If A and B are any two events, then )BA(P)B(P)A(P)BA(P ∩−+=∪ .

Corollary 1 If A and B are mutually exclusive, then )B(P)A(P)BA(P +=∪ .

Probability

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Corollary 2 If ,1A 2A , 3A , . . . , nA are mutually exclusive, then ∪∪ 21 AA(P . . . ++=∪ )A(P)A(P)An 21 . . . )A(P n+ . Theorem 2 If A and A’ are complementary events, then 1=+ )'A(P)A(P .

Example 13 In a college graduating class of 100 students, 54 studied mathematics, 69

studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that

a. the student took mathematics or history; b. the student did not take any of these courses;

c. the student took history but not mathematics.

Solution

Example 14 If A, B, and C are mutually exclusive events and P(A) = 0.2, P(B) = 0.3,

and P(C) = 0.4, find a. P(A ∪ B ∪ C) b. P[A’ ∩ (B ∪ C)] c. P(B ∪ C’)’

Solution

Example 15

The probability that an American industry will locate in Manila is 0.7, the probability that it will locate in Baguio is 0.4, and the probability that it will locate in either Manila or Baguio is 0.8. What is the probability that the industry will locate in both cities?

Solution

Probability

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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). The probability of B, given A, denoted by P(B|A), is defined by the equation

)A(P)BA(P

)A/B(P∩

= if P(A) > 0

Theorem 3 Two events A and B are independent if and only if P(B|A) = P(B) and P(A|B) = P(A) Otherwise, A and B are dependent. Theorem 4 (Multiplicative Rule) If in an experiment the events A and B can both occur, then P(A ∩ B) = P(A) • P(B|A) Theorem 5 If A and B are independent events, then P(A ∩ B) = P(A) • P(B) Theorem 6 (Generalized Multiplicative Rule) If, in an experiment, the events A1, A2, A3, . . . , Ak can occur, then P(A1 ∩ A2 ∩ A3 ∩ . . . ∩Ak) = P(A1) P(A2|A1) P(A3|A1 ∩ A2) . . . P(Ak| A1 ∩ A2 ∩ A3 ∩ . . . ∩ Ak-1) If the events A1, A2, A3, . . . , Ak are independent, then P(A1 ∩ A2 ∩ A3 ∩ . . . ∩Ak) = P(A1) P(A2) P(A3) . . . P(Ak) Example 16

If R is the event that a convict committed armed robbery and D is the event that the convict pushed dope, state in words what probabilities are expressed by

a. P(R|D) b. P(D’|R) c. P(R’|D’)

Solution

Probability

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Example 17

A random sample of 200 adults are classified below by sex and their level of education attained. If a person is picked at random from this group, find the probability that a. the person is a male, given that the person has secondary education; b. the person does not have a college degree , given that the person is a female.

Solution Example 18

The probability that an automobile being filled with gasoline will also need an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and filter need changing is 0.14. a. If the oil had to be changed, what is the probability that a new oil filter is needed? b. If a new oil filter is needed, what is the probability that the oil had to be changed?

Solution

Education Gender

Male Female Elementary 38 45 Secondary 28 50 College 22 17

Probability

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Example 19

The probability that a doctor correctly diagnoses a particular illness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient enters a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues?

Solution Example 20

A town has two fire engines operating independently. The probability that a specific fire engine is available when needed is 0.96.

a. What is the probability that neither engine is available when needed? b. What is the probability that a fire engine is available when needed?

Solution Example 21

Find the probability of selecting four good quarts of milk in succession from a cooler containing 20 quarts of which 5 have spoiled.

Solution Theorem 7 (Bayes’ Rule) If the events B1, B2, . . . , Bk constitute a partition of the sample space S where P(Bi) ≠ 0 for i= 1, 2, . . . , k, then for any event A in S such that P(A) ≠ 0, P(A) = P(B1 ∩ A) + P(B2 ∩ A) + . . . + P(Bk ∩ A) = P(B1) P(A|B1) + P(B2) P(A|B2) + . . . + P(Bk) P(A|Bk) P(Bi /A) =

)A(P)AB(P i I

Probability

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Example 22

In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78, and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06,

a. What is the probability that a person is diagnosed as having cancer?

b. What is the probability that a person diagnosed as having cancer actually has the disease?

Solution Example 23

Suppose that colored balls are distributed in three indistinguishable boxes as follows:

Boxes 1 2 3

Red White Blue

2 3 5

4 1 3

3 4 3

A box is selected at random from which a ball is drawn at random. a. Find the probability that the ball is red. b. Given that the ball is red, what is the probability that box 3 is selected?

Solution