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Statistics - Probability

This document provides study materials related to Statistics - Probability. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

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Study GuideStatistics–Probability1.Classic Theory of ProbabilityTheclassic theory of probabilityis one of the most common ways we understand probability instatistics.It explains probability using a simple idea:Probability = the number of ways something can happen ÷ the total number of possibleoutcomes1.1The Basic Probability Formula•Favorable outcomes= the outcomes youwant(also called “successes”)•Total outcomes=allpossible outcomesExample: Drawing an Ace from a Deck of CardsImagine you randomly pickone cardfrom a well-shuffled deck.•Total cards in a deck =52•Number of aces in a deck =4So, the probability of drawing an ace is:That means:So, the probability of getting an ace in one draw is0.077.

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Study Guide1.2Probability Values (Decimals)In statistics, probability is usually written as adecimal.It always falls between:•0= no chance at all•1= guaranteed (certainty)So probabilities range from0 to 1.1.3Important Assumption: All Outcomes Are Equally LikelyThe classic theory works only whenevery outcome has the same chance of happening.For example, in a fair deck of cards:•Every card has an equal chance of being chosen.•No card is more likely than another.That’s why this method works well for things like:•flipping fair coins•rolling fair dice•selecting from a well-shuffled deck of cards1.4Another Key Condition: Mutually Exclusive OutcomesThe classic theory applies to outcomes that aremutually exclusive (disjoint).Mutually exclusivemeans:Two outcomescannot happen at the same timein one trial.Example 1: Flipping a CoinIn one coin flip, you can get:•HeadORTail

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Study GuideBut you cannot get:•Head AND Tailat the same timeSo, head and tail aremutually exclusiveoutcomes.Example 2: Drawing One CardIf you draw one card, it can be:•anAceOR aKingBut it cannot be:•anAce AND a Kingin the same single drawSo, “ace” and “king” are alsomutually exclusiveoutcomes when drawing one card.Quick SummaryClassic probabilityuses this idea:It works best when:•All outcomes areequally likely•Outcomes aremutually exclusive(cannot happen together)2.Relative Frequency Theory of ProbabilityTherelative frequency theory of probabilityexplains probability usingreal results from repeatedtrials.Instead of assuming outcomes are equally likely (like the classic theory), this approach says:If you repeat an experimenta very large number of times, the percentage of times an outcomehappens becomes a good estimate of itsprobability.

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Study GuideThe Main IdeaThink of probability like a pattern you notice after doing something again and again:•The more times you repeat the experiment,•the closer theobserved percentagegets to thetrue probability.So basically:Probability ≈ (Number of times the outcome happens) ÷ (Total number of trials)Example: Faulty WidgetsSuppose a machine makes10,000 widgets(one at a time).•Total widgets produced =10,000•Faulty widgets =1,000So the probability that a widget is faulty is:That means the machine produces faulty widgets about10% of the time.Quick Summary•Relative frequency probabilityis based onactual repeated results•When the number of trials is extremely large,thepercentage of outcomesbecomes close to thereal probability3.Probability of Simple EventsAn easy probability example usingthree coins:ApennyAnickelAdime

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Study GuideEach coin can land either:•Head (H)or•Tail (T)Example 1: What is the probability that all three coins land on heads?To solve this, we use theclassic theory of probability:Step 1: Count the total outcomesSince each coin has2possible results (H or T), the total number of outcomes is:2×2×2=8So there are8 total possible outcomes.Step 2: Count the favorable outcomes“All three land heads” means:H H HThis happens inonly 1 outcome.Step 3: Find the probabilityFinal Answer:The probability is1/8, or0.125.

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Study GuideExample 2: What is the probability that exactly two of the three coins land heads?Step 1: Total outcomes stay the sameThere are still8 total outcomes.Step 2: Count outcomes with exactly two headsExactly two heads meanstwo H’s and one T.These possible outcomes are:•H H T•H T H•T H HThat gives us3 favorable outcomes.

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Study GuideStep 3: Find the probabilityFinal Answer:The probability is3/8, or0.375.Quick Summary•All three heads:•Exactly two heads:4.Quiz: Probability of Simple Events1. QuestionWhat is the probability that the number is3?Answer Choices• 1/10• 3/10• 1/3Correct Answer1/10Why This Is CorrectIf there are10 equally likely numbers(such as 1 to 10), onlyoneof them is3.

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Study GuideSo the probability is:1 out of 10 = 1/102. QuestionWhat is the probability that the number isgreater than 6?Answer Choices• 1/10• 3/10• 2/5Correct Answer2/5Why This Is CorrectNumbers greater than 6 (from 1 to 10) are:7, 8, 9, 10→4 numbersTotal possible outcomes =10Probability =4/10 = 2/53. QuestionWhat is the probability that the number isless than or equal to 3?Answer Choices• 1/10• 3/10• 2/5

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Study GuideCorrect Answer3/10Why This Is CorrectNumbers less than or equal to 3 are:1, 2, 3→3 numbersTotal outcomes =10Probability =3/104. QuestionWhat is the probability that the number isodd?Answer Choices• 1/10• 2/5• 1/2Correct Answer1/2Why This Is CorrectOdd numbers from 1 to 10 are:1, 3, 5, 7, 9→5 numbersTotal outcomes =10Probability =5/10 = 1/25. QuestionWhat is the probability that the number isnot 8?

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Study GuideAnswer Choices• 1/10• 1/2• 9/10Correct Answer9/10Why This Is CorrectOnlyone outcomeis 8.So outcomes that arenot 8=9 outcomesout of10Probability =9/106. QuestionWhat is the probability that the number is4 or 7?Answer Choices• 1/10• 1/5• 2/5Correct Answer1/5Why This Is CorrectFavorable outcomes:4 or 7→2 numbersTotal outcomes =10Probability =2/10 = 1/5
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Document Details

University
Texas A&M University
Course
Statistics
Subject
Statistics

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