Probability Basic Concepts PDF

PROBABI LITY The theory of probability grew out of the study of various games of chances using coins, dice and cards PROBABILITY BASIC CONCEPTS EVENT– processes such as flipping a coin, rolling a die or drawing a card from the deck, the performing of an experiment OUTCOME – the result of rolling t...

PROBABI LITY The theory of probability grew out of the study of various games of chances using coins, dice and cards PROBABILITY BASIC CONCEPTS EVENT– processes such as flipping a coin, rolling a die or drawing a card from the deck, the performing of an experiment OUTCOME – the result of rolling the dice, taking the measurement, etc SAMPLE SPACE – is a set of all possible outcomes of a probability experiment SAMPLE SPACES AND PROBABILITY Experiment Sample Space ______________________________________________ Toss one coin Head and Tail Roll a die 1,2,3,4,5,6 Answer a true/false question True , False Toss two coins HH, TT, HT, TH  SAMPLE SPACES AND PROBABILITY Example 1 Find the sample space for rolling 2 dice  SAMPLE SPACES AND PROBABILITY Example 2 DRAWING CARDS Find the sample space for drawing cards from ordinary deck of cards SAMPLE SPACES AND PROBABILITY Example 3 GENDER OF CHILDREN Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girls Solution There are two genders, male and female, and each child could be either gender. Hence, there are eight possibilities SAMPLE SPACES AND PROBABILITY CLASSICAL PROBABILITY Classical Probability uses samples to determine the numerical probability that an event will event will happen. It assumes that all the outcomes in sample space are equally likely to occur CLASSICAL PROBABILIT Y Example 5 Find the probability of getting a red ace when a card is drawn at random from an ordinary deck of cards CLASSICAL PROBABILIT Y Example 6 If the family has 3 children, find the probability that the two of the three are girls CLASSICAL PROBABILITY Example 7 A card is drawn from an ordinary deck. Find these probabilities a. Of getting of a jack b. Of getting 6 of clubs c. Of getting 3 or a diamond d. Of getting 3 or a 6 BASIC PROBABILITY RULE PROBABILITY RULE 1 The probability of any event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by 0 The rule states that probabilities cannot be negative or greater than 1 BASIC PROBABILITY RULE PROBABILITY RULE 2 If an event E cannot occur (i.e. the event contains no members in the sample space), its probability is 0 BASIC PROBABILITY RULE Example 8 When the single die is rolled, find the probability of getting a 9 Since the sample space is 1,2,3,4,5,6 and it is imposible to get 9 the probability is P(9) = BASIC PROBABILITY RULE PROBABILITY RULE 3 If an event E is certain, the probability of E is 1. BASIC PROBABILITY RULE Example 9 When the single die is rolled, find the probability of getting a number less than 7 Since all outcomes --- 1,2,3,4,5,6 are less than 7 P(7) = BASIC PROBABILITY RULE PROBABILITY RULE 3 The sum of the probability of all the outcomes in the sample space is 1 COMPLIMENTARY EVENTS When a die is rolled, for instance, the sample space consists of the outcomes 1,2,3,4,5 and 6. The event E of getting odd numbers consists of the outcomes 1,3 and 5. The event of not getting an odd number is called the compliment of event E, and it consists of the outcomes 2,4, and 6 COMPLIMENTARY EVENTS The complement of an event E is the set of outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by  COMPLIMENTARY a. Getting 1,2,3,5 and EVENTS Example 10 6 Find the complement of each b. Getting a event consonant a. Rolling a die and getting a 4 c. Getting Feb, Mar, b. Selecting a letter of the April, May, Aug, Sept, alphabet and getting a vowel Oct, Nov or Dec c. Selecting a month and d. Getting Saturday getting a month that begins with a J and Sunday d. Selecting a day of the week and getting a weekday  RULE FOR COMPLIMENTARY EVENTS COMPLIMENTARY EVENTS EMPIRICAL PROBABILITY Classical Probability assumes that the certain outcomes are equally likely (such as the outcomes when a die is rolled) Empirical Probability relies on the actual experience to determine the likelihood of outcomes EMPIRICAL PROBABILITY The researcher asked 50 people who plan to travel over Christmas holiday how they will get to their destination. The results can be categorized in a frequency distribution as shown. Method Frequency Drive 41 Fly 6 Train or bus 3 50 EMPIRICAL PROBABILITY  EMPIRICAL PROBABLITY Example 12 In the travel survey P(E) = = = just described, find the probability that a person will travel by airplane over the Christmas holiday  EMPIRICAL a. A person has type O blood PROBABALITY b. A person has type A Example 13 Distribution of or type B blood Blood Type c. A person has neither In a sample of 50 people, 21 type A or type O blood had type O blood, 22 had type A blood, 5 had type B d. A person does not blood and 2 had type AB have type AB blood blood. Set up a frequency distribution and find the following probabilities a. A person has type O EMPIRICAL blood PROBABILITY b. A person has type A Example 13 Distribution of or type B blood Blood Type c. A person has neither Type Frequency type A or type O blood A 22 B 5 d. A person does not AB 2 have blood type AB 0 21 50 ADDITION RULE FOR PROBABILITY  ADDITION RULE FOR PROBABILITY EXAMPLE  THE EVENTS OF GETTING A 4 AND 6 WHEN A SINGLE CARD IS DRAWN FROM A DECK, SINCE A SINGLE CARD CANNOT BE BOTH A 4 AND A 6 HOW ABOUT GETTING 4 AND GETTING A HEART? NOT MUTUALLY EXCLUSIVE. SINCE YOU CAN SELECT THE 4 OF HEART WHEN DRAWING A SINGLE CARD FROM AN ORDINARY DECK.  ADDITION RULE 1 Example A1 Selecting Doughnut A box contains 3 original a.P(glazed or oreo Kreme) = P(glazed) + glazed, 4 new York P(oreo) cheesecake, and 5 oreo Kreme.If a person selects a P= doughnut at random, find the probability that is either original glazed or oreo Kreme.  ADDITION RULE 1 Example A2 Selecting a day in a Week a.P(Saturday or A day of the week is Sunday) = P(Saturday) selected at random. + P(Sunday) Find the probability P= that is a weekend day. ADDITION RULE FOR PROBABILITY  ADDITION RULE 2 a.P(ace or black card) Example A2.1 Drawing a Card = P(ace) + P(black card) – P(black aces) A single card is drawn at random from an ordinary deck of cards. Find the probability that it is either an ace or a black P= card.  ADDITION RULE 2 Example A2.2 Selecting a Medical Staff Person a. P(nurse or male) = In a hospital unit there are P(nurse) + P(male) – P(male 8 nurses and 5 physicians; nurse) 7 nurses and 3 physicians P= are females. If a staff person is selected, find the probability that the subject is a nurse or a male. MULTIPLICATION RULES MULTIPLICATION RULES CAN BE USED TO FIND THE PROBABILITY WHEN TWO OR MORE EVENTS THAT OCCUR IN SEQUENCE. MULTIPLICATION RULE FOR PROBABILITY  MULTIPLICATON a.P(queen and ace) = RULE 1 Example M1.1 Drawing a Card P(queen) P(ace) A card is drawn from a deck and replaced; then the second card is drawn. Find the probability of getting a queen and then P= an ace.  MULTIPLICATION a.P(head and 4) = RULE 1 Example M1.2 Drawing a Card P(head) P(4) P= A coin is flipped and a die is rolled. Find the probability of getting a head and a 4 on a die  MULTIPLICATION a. Selecting 2 blue RULE 1 Example M1.3 Selecting a balls Colored Ball b. Selecting 1 blue ball and then 1 white A box contains 3 red balls, 2 ball blue balls, and 5 white balls. c. Selecting 1 red ball A ball is selected and its and then 1 blue ball colored noted. Then it is a. replaced. A second ball is b. selected and it is color noted. Find the probability c. of each of these: MULTIPLICATION RULE FOR PROBABILITY Suppose a card is drawn from a deck and not replaced , and then a second card is drawn. What is the probability of selecting an ace on the first card and king on the second card. P(ace) = P(king) = P(ace and king) = = MULTIPLICATION RULE FOR PROBABILITY The event of getting a king on the second draw given that the ace was drawn the first time is called the conditional probability The conditional probability of an event B in a relationship to an event A is the probability that event B occurs after event A has already occurred. The MULTIPLICATION RULE FOR PROBABILITY  WHAT ARE DEPENDENT EVENTS  Two events are dependent if the result of the first event affects the probability of the second happening  In other words, the second event depends on what happened in the first event.  Example:  Suppose you have a bag with 3 red balls and 2 blue balls  You pick one ball and do not put it back  Now if you pick a second ball, the total number of balls has changed, so the probability is affected.  The first draw affects the chances of the second draw, so these events are dependent MULTIPLICATION RULE FOR PROBABILITY Here are examples of dependent events Drawing a card from a deck, not replacing it, and then drawing a second card Selecting a ball from an urn, not replacing it, and then selecting a second ball Having high grades and getting scholarship Parking in a no-parking zone and getting a parking ticket MULTIPLICATION In this case the events RULE 2 Example M2.1 University Crime are dependent since the researcher wishes At a university in Recto, there to investigate two were 5 bulgaries reported in distinct cases. The first 2003, 16 in 2004, and 32 in case selected and not 2005. If a researcher wishes to replaced. select a random two bulgaries to further investigate, find the probability that both will have P(and)=P() P( occurred in 2004. = MULTIPLICATION RULE 2 MULTIPLICATION RULE 2