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Statistics - Probability

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    Study Guide Statistics – Probability 1. Classic Theory of Probability The classic theory of probability is one of the most common ways we understand probability in statistics. It explains probability using a simple idea: Probability = the number of ways something can happen ÷ the total number of possible outcomes 1.1 The Basic Probability Formula • Favorable outcomes = the outcomes you want (also called “successes”) • Total outcomes = all possible outcomes Example: Drawing an Ace from a Deck of Cards Imagine you randomly pick one card from a well - shuffled deck. • Total cards in a deck = 52 • Number of aces in a deck = 4 So, the probability of drawing an ace is: That means: So, the probability of getting an ace in one draw is 0.077 .

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    Study Guide 1.2 Probability Values (Decimals) In statistics, probability is usually written as a decimal . It always falls between: • 0 = no chance at all • 1 = guaranteed (certainty) So probabilities range from 0 to 1 . 1.3 Important Assumption: All Outcomes Are Equally Likely The classic theory works only when every 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 cards 1.4 Another Key Condition: Mutually Exclusive Outcomes The classic theory applies to outcomes that are mutually exclusive (disjoint) . Mutually exclusive means: Two outcomes cannot happen at the same time in one trial. Example 1: Flipping a Coin In one coin flip, you can get: • Head OR Tail

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    Study Guide But you cannot get: • Head AND Tail at the same time So, head and tail are mutually exclusive outcomes. Example 2: Drawing One Card If you draw one card, it can be: • an Ace OR a King But it cannot be: • an Ace AND a King in the same single draw So, “ace” and “king” are also mutually exclusive outcomes when drawing one card. Quick Summary Classic probability uses this idea: It works best when: • All outcomes are equally likely • Outcomes are mutually exclusive (cannot happen together) 2. Relative Frequency Theory of Probability The relative frequency theory of probability explains probability using real results from repeated trials . Instead of assuming outcomes are equally likely (like the classic theory), this approach says: If you repeat an experiment a very large number of times , the percentage of times an outcome happens becomes a good estimate of its probability .

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    Study Guide The Main Idea Think of probability like a pattern you notice after doing something again and again: • The more times you repeat the experiment, • the closer the observed percentage gets to the true probability . So basically: Probability ≈ (Number of times the outcome happens) ÷ (Total number of trials) Example: Faulty Widgets Suppose a machine makes 10,000 widgets (one at a time). • Total widgets produced = 10,000 • Faulty widgets = 1,000 So the probability that a widget is faulty is: That means the machine produces faulty widgets about 10% of the time . Quick Summary • Relative frequency probability is based on actual repeated results • When the number of trials is extremely large, the percentage of outcomes becomes close to the real probability 3. Probability of Simple Events A n easy probability example using three coins : A penny A nickel A dime

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    Study Guide Each 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 the classic theory of probability : Step 1: Count the total outcomes Since each coin has 2 possible results (H or T), the total number of outcomes is: 2 × 2 × 2 = 8 So there are 8 total possible outcomes . Step 2: Count the favorable outcomes “All three land heads” means: H H H This happens in only 1 outcome . Step 3: Find the probability Final Answer: The probability is 1/8 , or 0.125 .

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    Study Guide Example 2: What is the probability that exactly two of the three coins land heads? Step 1: Total outcomes stay the same There are still 8 total outcomes . Step 2: Count outcomes with exactly two heads Exactly two heads means two H’s and one T . These possible outcomes are: • H H T • H T H • T H H That gives us 3 favorable outcomes .

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    Study Guide Step 3: Find the probability Final Answer: The probability is 3/8 , or 0.375 . Quick Summary • All three heads: • Exactly two heads: 4. Quiz: Probability of Simple Events 1. Question What is the probability that the number is 3 ? Answer Choices • 1/10 • 3/10 • 1/3 Correct Answer 1/10 Why This Is Correct If there are 10 equally likely numbers (such as 1 to 10), only one of them is 3 .

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    Study Guide So the probability is: 1 out of 10 = 1/10 2. Question What is the probability that the number is greater than 6 ? Answer Choices • 1/10 • 3/10 • 2/5 Correct Answer 2/5 Why This Is Correct Numbers greater than 6 (from 1 to 10) are: 7, 8, 9, 10 → 4 numbers Total possible outcomes = 10 Probability = 4/10 = 2/5 3. Question What is the probability that the number is less than or equal to 3 ? Answer Choices • 1/10 • 3/10 • 2/5

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    Study Guide Correct Answer 3/10 Why This Is Correct Numbers less than or equal to 3 are: 1, 2, 3 → 3 numbers Total outcomes = 10 Probability = 3/10 4. Question What is the probability that the number is odd ? Answer Choices • 1/10 • 2/5 • 1/2 Correct Answer 1/2 Why This Is Correct Odd numbers from 1 to 10 are: 1, 3, 5, 7, 9 → 5 numbers Total outcomes = 10 Probability = 5/10 = 1/2 5. Question What is the probability that the number is not 8 ?

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    Study Guide Answer Choices • 1/10 • 1/2 • 9/10 Correct Answer 9/10 Why This Is Correct Only one outcome is 8. So outcomes that are not 8 = 9 outcomes out of 10 Probability = 9/10 6. Question What is the probability that the number is 4 or 7 ? Answer Choices • 1/10 • 1/5 • 2/5 Correct Answer 1/5 Why This Is Correct Favorable outcomes: 4 or 7 → 2 numbers Total outcomes = 10 Probability = 2/10 = 1/5

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