CHAPTER 12: PROBABILITY AND STATISTICS

CHAPTER 12: PROBABILITY AND STATISTICS Slide 1

Section 12-1THE COUNTING PRINCIPLE Thursday 5-11-17 Definitions: • Outcome-the result of a single trial. Ex. Flipping a coin has two outcomes (head or tail). • Sample Space-the set of all possible outcomes. • Event-consists of one or more outcomes of a trial. • Independent Events-two events are independent if the occurrence (or non-occurrence) of one of the events does not affect the probability of the occurrence of the other event. Ex. The choices of letters and digits to be put on a license plate, because each letter or digit chosen does not affect the choices for the others.

12-1 Example Probability experiment: Roll a die Sample space: { 1 2 3 4 5 6 } Event: { Die is even } = { 2 4 6 } Outcome: {3}

Section 12-1THE COUNTING PRINCIPLE 5) Fundamental Counting Principle-If there are “a” ways for one activity to occur and “b” ways for a second activity to occur, then there are (a × b) ways for both to occur. Ex. There are 6 ways to roll a die and 2 ways to flip a coin… 6 × 2 = 12 ways for both to occur. 6) Dependent events-the outcome of one event DOES affect the out come of another event.

12-1: Example: When two coins are tossed, what is the probability that both are tails? Tree Diagram First Coin H T / \ / \ Second Coin H T H T | | | | Possible HH HT TH TT Outcomes 1 2 3 4 Therefore, 4 possible outcomes. Fundamental Counting Principle: 2 possible outcomes first coin X 2 possible outcomes second coin = 4 possible outcomes P(both tails)= 1 4

12-1: Independent and Dependent Events • Independent events: if one event occurs, it does not affect the probability of the other event • Drawing cards from two decks • Dependent events: if one event affects the outcome of the second event, changing the probability • Drawing two cards in succession from same deck without replacement

12-1: Examples Independent Events A = Being female B = Having type O blood A = 1st child is a boy B = 2nd child is a boy Dependent Events A = taking an aspirin each day B = having a heart attack A = being a female B = being under 64” tall

12-1: Examples • Determine if the following events are independent or dependent. • 1. 12 cars are on a production line where 5 are defective and 2 cars are selected at random. • A = first car is defective • B = second car is defective. • Two dice are rolled. • A = first is a 4 and B = second is a 4 Skills Practice Workbook pg. 79 (#1-11) SG&I workbook pg. 157 (#1-8)

12-3: PROBABILITY Friday 5-12-17 Definition: -Probability is the measure of how likely something will occur. -It is the ratio of desired outcomes to total outcomes. PROBABILITY= #desired #total -Probabilities MUST be simplified if possible.

12-3: EXAMPLE Flip a coin… -What is the probability you get heads? -What is the probability you get tails? (Remember think of all the possible outcomes)

12-3: Tree Diagrams Two six-sided dice are rolled. Describe the sample space. Start 1st roll 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 2nd roll 6 possible outcomes on die #1 X 6 possible outcomes on die #2 = 36outcomes

12-3: Sample Space and Probabilities Two dice are rolled and the sum is noted. 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Find the probability the sum is 4. 3/36 = 1/12 = 0.083 Find the probability the sum is 11. 2/36 = 1/18 = 0.056 5/36 = 0.139 Find the probability the sum is 4 or 11.

12-3: Probability of Success & Failure Definitions: 1-Success-desired outcome 2-Failure-any other outcome 3-Random-when all outcomes have an equally likely chance of occurring 4-Odds-another way to measure the chance of an event occurring, it is the ratio of the number of successes to the number of failures. Ex.Success:failure

12-3: PROBABILITY VS. ODDS Ex. What is the probability of rolling a die and getting the number 5? • Probability = #desiredProbability=1 total # 6 Now what are the odds? • ODDS = success:failure ODDS= 1:5 (means one chance of getting a number 5 : five chances of not getting a number 5.)

12-3: Odds • Odds can be written as a fraction or ratio (but must be simplified just like probabilities): • Success Failures Success:Failures Note: In sports we often look at wins over losses.

12-3: Converting from Probability to ODDS EXAMPLE: Flipped a coin four times… P(two tails)= ¼ = .25 or 25% chance What are the odds? ODDS= 1 success 3 failures 1:3

12-3: PROBABILITY DISTRIBUTIONS Monday 5-15-17 Definitions: 1-random variable-a variable whose value is the numerical outcome of a random event. Ex. When rolling a die we can let the random variable “D” represent the number showing on the die. So, “D” can equal 1, 2, 3, 4, 5, or 6.

12-3: PROBABILITY DISTRIBUTIONS 2-probability distribution-for a particular random variable it is a function that maps the sample space to the probabilities of the outcomes in the sample space. Ex.

12-3: PROBABILITY DISTRIBUTIONS 3-uniform distribution-a distribution where all of the probabilities are the same. 4-relative-frequency histogram-a table of probabilities or graph that helps to visualize a probability distribution.

12-3: Probability Probability is based on observations or experiments. THERE ARE TWO TYPES OF PROBABILITY: 1- Experimental AND 2-Theoretical Definitions: An experimental probability is one that happens as the result of an experiment. # outcomes # trials ***The probability we have done so far has been “theoretical probabilities,” because there was no experiment. **Except when you do a project!

12-3: THEORETICAL AND EXPERIMENTAL PROBABILITY The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur. • If P(E) = 0, then event E is impossible. • If P(E) = 1, then event E is certain. 0  P(E)  1 Impossible Even Certain 0 .5 1

12-3: THEORETICAL AND EXPERIMENTAL PROBABILITY THE THEORETICAL PROBABILITY OF AN EVENT 4 P (A) = 9 total number of outcomes all possible outcomes You can express a probability as a fraction, a decimal, or a percent.For example: , 0.5, or 50%. 1 2 The theoretical probability of an event is often simply called the probability of the event. When all outcomes are equally likely, the theoretical probability that an event Awill occur is: number of outcomes in A P (A) = outcomes in event A

12-3: EXAMPLE Theoretical P(A) = number if ways A can occur total number of outcomes In a bag you have 3 red marbles, 2 blue marbles and 7 yellow marbles. If you select one marble at random, P(red) =

12-3: EXAMPLE Theoretical P(A) = number if ways A can occur total number of outcomes In a bag you have 3 red marbles, 2 blue marbles and 7 yellow marbles. If you select one marble at random, P(red) = 3 / (3+2+7) = 3/12 = 1/4

12-3: Experimental examples • Prentice went fishing at a pond that contains three types of fish: blue gills, red gills and crappies. He caught 40 fish and recorded the type. The following frequency distribution shows his results. Fish Type Number of times caught Blue gill 13 Red gill 17 Crappy 10 If you catch a fish, what is the probability that it is a blue gill? A red gill? A crappy?

12-3: Subjective Probability • Subjective probability results from educated guesses, intuition and estimates. • Examples… • A doctor’s prediction that a patient has a 90% chance of full recovery • A business analyst predicting an employee strike being 0.25

12-3: Summary • Classical (Theoretical) • The number of outcomes in a sample space is known and each outcome is equally likely to occur. • Empirical (Statistical) • A.K.A. Experimental • The frequency of outcomes in the sample space is estimated from experimentation. • Subjective (Intuition) • Probabilities result from intuition, educated guesses, and estimates. • COMPLETE PROBABILITY AND ODDS WORKSHEET

12-4: Independent Events Tuesday 5-16-17 Whatever happens in one event has absolutely nothing to do with what will happen next because: • The two events are unrelated OR • You repeat an event with an item whose numbers will not change (ex. spinners or dice) OR • You repeat the same activity, but you REPLACE the item that was removed. The probability of two independent events, A and B, is equal to the probability of event A times the probability of event B. P(A∩B) = P(A) ● P(B) Slide 28

12-4: Multiplication Rule for Independent Events • To get probability of both events occurring, multiply probabilities of individual events • Ace from first deck and spade from second • Probability of ace is 4/52 = 1/13 • Probability of spade is 13/52 = 1/4 • Probability of both is 1/13 x 1/4 = 1/52

12-4: INDEPENDENT EVENTS Practice: Roll a die and flip a coin: 1-P(heads and 6) = 2-P(tails and a 5) =

12-4: INDEPENDENT EVENTS Practice: Roll a die and flip a coin: 1-P(heads and 6) = ½ x 1/6 = 2-P(tails and a 5) =

12-4: INDEPENDENT EVENTS Practice: Roll a die and flip a coin: 1-P(heads and 6) = ½ x 1/6 = 1/12 2-P(tails and a 5) =

12-4: INDEPENDENT EVENTS Practice: Roll a die and flip a coin: 1-P(heads and 6) = ½ x 1/6 = 1/12 2-P(tails and a 5) = ½ x 1/6 =

12-4: INDEPENDENT EVENTS Practice: Roll a die and flip a coin: 1-P(heads and 6) = ½ x 1/6 = 1/12 2-P(tails and a 5) = ½ x 1/6 = 1/12

P S O T R 6 1 5 2 3 4 12-4: Independent Events Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? P(even) = P(vowel) = P(even and vowel) = Slide 35

P S O T R 6 1 5 2 3 4 12-4: Independent Events Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? P(even) = (3 evens out of 6 outcomes) (1 vowel out of 5 outcomes) P(vowel) = P(even and vowel) = Slide 36

P S O T R 6 1 5 2 3 4 12-4: Independent Events Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? P(even) = (3 evens out of 6 outcomes) (1 vowel out of 5 outcomes) P(vowel) = P(even and vowel) = Slide 37

12-4: Independent Events Find the probability P(jack and factor of 12) x = Slide 38

12-4: Independent Events Find the probability • P(6 and not 5) x = Slide 41

12-4: Dependent Event • What happens during the second event depends upon what happened before. • In other words, the result of the second event will change because of what happened first. • Determining the probability of a dependent event is usually more complicated than finding the probability of a independent event. The probability of two dependent events, A and B, is equal to the probability of event A times the probability of event B. However, the probability of event B now depends on event A. P(A∩B) = P(A) ● P(B) Slide 44

12-4: So for both INDEPENDENT & DEPENDENT EVENTS… To find the probability that two events, A and B will occur in sequence, multiply the probability A occurs by the conditional probability B occurs, given A has occurred. P(A and B) = P(A) x P(B ) ***EXCEPTION: For dependent, P(B given A) DEPENDENT EXAMPLE: Two cars are selected from a production line of 12 where 5 are defective. Find the probability both cars are defective. A = first car is defective B = second car is defective. P(A) = 5/12 P(B given A) = 4/11 P(A and B) = 5/12 x 4/11 = 5/33 = 0.1515

12-4: Probability Examples Ex.1) Independent Events: Spinner #1 is partitioned into three equal sections, colored black, white, and grey. Spinner #2 is partitioned into four equal sections, colored red, blue, green, and yellow. If both spinners are spun, what is the probability of getting black and red?

12-4: Probability Example Since we expect to get black one-third of the time, and we expect to get red one-quarter of the time, then we expect to get black one-third and red one-quarter of the time. . . Imagine a tree diagram where the first column shows the three outcomes for Spinner #1, each of which is followed by the four outcomes for Spinner #2 in the second column. Three groups of four branches creates 12 possible outcomes.

12-4: Probability Example Imagine a tree diagram where the first column shows the three outcomes for Spinner #1, each of which is followed by the four outcomes for Spinner #2 in the second column. Three groups of four branches creates 12 possible outcomes.

12-4: Probability Example Ex.2) Dependent Events: A bag contains 10 marbles; 5 red, 3 blue, and 2 silver. If you draw one marble at random and hold it in your left hand, and then draw a second marble at random and hold it in your right hand, what is the probability that you are holding two silver marbles? It’s easy to determine the probability of the first marble being silver. However, notice that if you start by getting a silver marble and then try for the second, the bag will be different. How? Now, there is only one silver marble in a bag containing a total of 9 marbles. . .

12-4: Probability Example

CHAPTER 12: PROBABILITY AND STATISTICS