PROBABILITY (Introduction)

Random Experiments

Sample Spaces

Events

Random Experiment (or Experiment)

A random experiment is any process or activity that generates a set of data. It is repeatable under basically the same condition, leading to well-defined outcomes. It is random because we can never tell in advance what the realization is going to be even if we can specify what the possible outcomes are. If we think an experiment as being performed repeatedly, each repetition is called a trial. We observe an outcome for each trial.

Examples of a random experiment:

1.Tossing an ordinary coin.2.Rolling a fair die.3.Randomly selecting a card from a deck.4.Recording the number or cars pulling up at a service station.5.Recording the yield of a new variety of rice in kg/ha.6.Finding out the gender of the first two children of families with at least two children in a barangay in Dasmariñas, Cavite

Sample Space

Sample space is the set of all possible outcomes of a random experiment. It is denoted by the Greek letter omega (Ω) or S. This is also known as the universal set.

Types of Sample Spaces

1.A finite sample space is a sample space with finite number of possible outcomes.2.An infinite sample space is a sample with infinite number of possible outcomes.

Natures of Sample Spaces

1.Discrete sample space is a sample space with a countable (finite or infinite) number or possible outcomes.2.Continuous sample space is a sample space with a continuum possible outcomes.

Event

An event is a subset of the sample space denoted by any letter of the English alphabet. An event is an outcome of a random experiment.

Types of Events

1.Elementary event – an event consisting of one possible outcome.2.Impossible event – an event consisting of no outcome and is denoted by { } or Ø.3.Sure event – an event consisting of all possible outcomes.4.Complement of an event – is the set of all elements of the sample space which not in the event A; denoted by Ac or A’.

Operations on Events

1.The intersection of 2 events A and B, denoted by A ∩ B, is the event containing all elements that are common to events A and B.

Note: Two events A and B are mutually exclusive if they cannot both occur simultaneously. That is, A ∩ B = { }

2.The union of 2 events A and B, denoted by A U B, is the set containing all the elements that belong to A or B or both.

3.Other operations:

A'A'

S'S'

SA'AA'A

AASSA

Exercise:

1.Set up the sample space for the single toss of a pair of dice and list the elements of the following events.a)A = event of obtaining a sum of 7 or 11b)B = event of obtaining a sum of at least 10c)C = event of obtaining a sum of at most 6d)D = event of obtaining a product of 24e)E = event of obtaining a 3 in exactly one of the dicef)F = event of obtaining a 3 on either dieg)G = event of obtaining a sum of 7 and a product of 12

PROBABILITY(Basic Concepts and Properties)

Probability

Probability is the chance that something will happen. Probabilities are expressed mathematically as fractions (1/6, 1/3, 8/9) or as decimals (.25, .5, .78) between 0 and 1. Assigning zero means that something can never happen; and a probability of 1 indicates that something will surely happen.

0

space sample theis where1

event any for 10

PP

AAP

Classical Probability

If an experiment can result in any one of N different equally likely outcomes, and if n of theses outcomes corresponds to event A, then the probability of event A is

N

nAP

Example 1:

If a card is drawn from an ordinary deck, find the probability that it is a heart.

Example 2:

A coin is tossed thrice. What is the probability that at least 2 heads occur?

Example 3:

A pair of dice is rolled. What is the probability that the sum is equal to: (a) 5, (b) 10, (c) at most 9, (d) at least 8.

Probability of a Complement

This rule states that the probability that an event A will not occur is equal to 1 minus the probability that it will occur.

APAP 1'

Example 1:

If the probability that a patient survives the operation is 3/5, what is the probability that he will not survive?

Example 2:

A sample of 586 adults were asked if they like the taste of pomelo juice and 235 positively respond. (a) Find the probability that an adult interviewed at random likes pomelo juice. (b) Find the probability that an adult does not like the taste of pomelo juice.

Probability Mutually Exclusive Events

If A and B are mutually exclusive events, then

BPAPBAP

Example:

Consider the experiment of drawing a card on a deck of 52 playing cards. What is the probability of drawing a face card or an ace card?

Probability Non-Mutually Exclusive Events

If A and B are non-mutually exclusive events, then

BAPBPAPBAP

Example:

Consider the experiment of drawing a card on a deck of 52 playing cards. What is the probability of drawing a red card or a face card?

Example 1:

If a card is drawn at random from an ordinary deck of 52 cards, find the probability that it is: (a) diamond, (b) a red card, (c) a heart or a spade, (d) not an ace, (e) a black card or a king.

Example 2:

The letters of the word ENGINEERING are written on slips of paper, and are placed in a box. A slip of paper is chosen at random. What is the probability that: (a) the letter is vowel, (b) the letter is a consonant.

Example 3:

Forty percent of the sales force at a large insurance company have laptop computers, 65 percent have desktop computers, and 24 percent have both. What percent of the sales people have either laptop or desktop computers.

Exercise 1:

How many 3 digit numbers greater than 300 can be formed from 0, 1, 2, 3, 4, 5 and 6 if each digit can be used only once?

Exercise 2:

How many sample points or elements are there when simultaneously a coin is tossed once, two dice are thrown and a card is selected at random from an ordinary deck of 52 cards?

Exercise 3:

From 7 consonants and 5 vowels, how many words consisting of 3 different consonants and 3 different vowels may be formed? The words need not have meaning?

Exercise 4:

A girl has 7 coins of different denominations. How many different sums of money can she form using 2 coins at a time?

Exercise 5:

If a quiz contains 5 true-false questions and 5 multiple-choice questions, each with 4 alternatives, in how many different ways can one answer the 10 questions?

Exercise 5:

How many ways can 6 passengers be lined up to get on a train: (a) if a certain 3 persons insist on following each other? If two persons refuse to follow each other?

Exercise 5:

How many ways can 6 passengers be lined up to get on a train: (a) if a certain 3 persons insist on following each other? If two persons refuse to follow each other?

Exercise 5:

A carton of 12 transistor batteries includes one that is defective. In how many different ways can an inspector choose three of the batteries and: (a) get one that is defective (b) not get the one that is defective.

Exercise 5:

Four married couples have bought eight seats in a row for a football game. In how many different ways can they be seated if: (a) each husband is to sit to the left of his wife? (b) all men are to sit together and all women are to sit together?

Exercise 5:

Four married couples have bought eight sets in a row for a football game. In how many different ways can they be seated if: (a) each husband is to sit to the left of his wife? (b) all men are to sit together and all women are to sit together?

Exercise 6:

A bowl contains 15 red beads, 30 white beads, 20 blue beads, and 7 black beads. If one of the beads is drawn at random, what are the probabilities that it will be: (a) red, (b) white or blue, (c) black, (d) neither white or black, (e) blue and red.