3.1.3 - Examples
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Classical Definition of Probability
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Today we're discussing the Classical Definition of Probability. Remember, this definition assumes all outcomes are equally likely. Can anyone tell me the formula for calculating probability?
Is it the number of favorable outcomes divided by the total outcomes?
Exactly right! Now, let's consider our first example: tossing a fair die. What are the total outcomes?
There are 6 possible outcomes.
Correct! If we want to find the probability of rolling an even number, how many favorable outcomes do we have?
Three: 2, 4, and 6.
Great! So, using the formula, what's P(E) for rolling an even number?
P(E) is 0.5.
Correct! Let’s summarize: For a fair die, the probability of an even number is 3 out of 6, or 0.5.
Example of Drawing a Card
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Now let's explore another example: drawing a card from a standard deck. What is the total number of outcomes here?
It’s 52 cards in a deck.
Exactly! If we want to draw a heart, how many favorable outcomes do we have?
There are 13 hearts.
Right! So, what’s the probability of drawing a heart?
That would be 0.25.
Correct! So we see that drawing from a card deck demonstrates the Classical Definition well.
Axiomatic Definition of Probability
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Now let’s switch gears to the Axiomatic Definition introduced by Kolmogorov. Has anyone heard about the sample space?
Is it all the possible outcomes of an experiment?
Exactly! In a fair coin toss, our sample space S is {H, T}. Can you define the collection of events in this case?
The events would be the subsets including the empty set, heads, tails, and both.
Perfect! Now, can someone assign probabilities to those events following the axioms?
P(H) = 0.5 and P(T) = 0.5.
Right! And how do these probabilities satisfy Kolmogorov’s axioms?
They are non-negative, add up to 1, and are mutually exclusive.
Excellent summary! Remember, this rigorous structure is what makes the Axiomatic definition so robust.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore practical examples, such as tossing a die and drawing from a card deck, to demonstrate the Classical Definition of Probability. Furthermore, we examine a coin toss scenario to illustrate the Axiomatic Definition, highlighting applications of probability in various contexts.
Detailed
Examples in Probability Theory
In this section, we provide concrete examples to elucidate the Classical and Axiomatic definitions of probability. Understanding these examples is essential as they lay the groundwork for future probability applications, especially in fields like engineering, where uncertainty must be quantified.
Classical Definition Examples
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Example 1: Tossing a Fair Die
- The total number of outcomes when tossing a fair die is 6: (1, 2, 3, 4, 5, 6).
- The event of interest (E) is rolling an even number, which has 3 favorable outcomes: (2, 4, 6).
- The probability of event E is calculated as:
$$P(E) = \frac{m}{n} = \frac{3}{6} = 0.5$$
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Example 2: Drawing a Card from a Standard Deck
- The total number of outcomes from a standard deck of 52 cards is 52.
- If we want to draw a heart, the number of favorable outcomes is 13.
- Therefore, the probability of drawing a heart is:
$$P(heart) = \frac{13}{52} = 0.25$$
Axiomatic Definition Example
- Example: Tossing a Fair Coin
- For a fair coin, we define the sample space (S) as {H, T}.
- The collection of events (F) includes all possible subsets: {∅, {H}, {T}, {H, T}}.
- The probability function assigns:
- P({H}) = 0.5
- P({T}) = 0.5
- This satisfies Kolmogorov’s three axioms:
- Non-negativity: P({H}) ≥ 0
- Normalization: P({H, T}) = 0.5 + 0.5 = 1
- Additivity: P({H} ∪ {T}) = P({H}) + P({T})
Understanding these examples provides clarity on how probabilities are calculated in practical scenarios, aiding students in grasping these pivotal concepts in probability theory.
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Example 1: Tossing a Fair Die
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Chapter Content
Example 1: Tossing a fair die
Total outcomes = 6 (1, 2, 3, 4, 5, 6) Favorable outcomes for getting an even number = 3 (2, 4, 6)
𝑃(even number) = = 0.5
6
Detailed Explanation
In this example, we are tossing a fair die, which is a common method to illustrate probability. A fair die has 6 sides, representing the numbers 1 through 6. When we talk about total outcomes, we mean all the possible results we could get from a single roll of the die, which are the six numbers: 1, 2, 3, 4, 5, and 6. The favorable outcomes for our specific event, which is rolling an even number, are 2, 4, and 6. Therefore, we have 3 favorable outcomes out of a total of 6. To find the probability of rolling an even number, we use the formula for probability: P(E) = Number of favorable outcomes / Total number of outcomes. Plugging in our numbers gives us P(even number) = 3/6 = 0.5.
Examples & Analogies
Think of rolling a die like flipping a coin, but with the coin having six sides instead of two. Whenever you toss the coin (or die), there is a chance for each side (or number) to land face up. Just as we have a 50% (0.5) chance to get heads or tails with a coin, we have a 50% chance to roll an even number with a die because 3 out of the 6 numbers are even.
Example 2: Drawing a Card from a Standard Deck
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Chapter Content
Example 2: Drawing a card from a standard deck
Total outcomes = 52 Favorable outcomes for drawing a heart = 13
𝑃(heart) = = 0.25
52
Detailed Explanation
In this example, we are drawing a card from a standard deck of 52 playing cards. The total number of outcomes here is 52 because there are 52 distinct cards in the deck. Now, among these, we are interested in finding the probability of drawing a heart. There are 13 hearts in a deck (the heart suit consists of Ace through King). So, the number of favorable outcomes for this event is 13. To find the probability of drawing a heart, we again use the probability formula: P(E) = Number of favorable outcomes / Total number of outcomes. Thus, we calculate P(heart) = 13/52, which simplifies to 0.25.
Examples & Analogies
Imagine you are at a party where there is a big bag filled with 52 colored cards, and you want to know how likely you are to pull out a red card (the hearts being red). Since there are 13 red cards out of 52 total cards, you can think of it like a game of chance: when you reach into the bag, your chances of grabbing a red card is 25%, similar to how you would have a 25% chance to roll an even number with a die.
Key Concepts
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Probability: A measure of the likelihood of an event occurring.
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Equally Likely Outcomes: Outcomes that have the same probability of occurring.
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Sample Space: The set of all possible outcomes of a random experiment.
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Axioms of Probability: Foundational principles that define the properties of probability functions.
Examples & Applications
Example of rolling a fair die to calculate the probability of getting an even number.
Example of drawing a card from a standard deck to find the probability of drawing a heart.
Example of tossing a fair coin and applying Kolmogorov's axioms.
Memory Aids
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Rhymes
When rolling the die and asking 'why?', count those evens, give it a try. Three out of six makes it so clear, the probability's half, let's give a cheer!
Stories
Imagine a magician with a deck of cards: 52 in total. He draws one heart and says, 'Ah! 13 ways to find a heart, that’s true magic in the art!'
Memory Tools
To remember Kolmogorov's axioms, use 'N-N-A' for Non-negativity, Normalization, Additivity.
Acronyms
For events in S, think FOLD for 'Favorable Outcomes / Total Outcomes' and that’s how it flows.
Flash Cards
Glossary
- Classical Definition of Probability
A foundational interpretation stating that if an experiment has n equally likely outcomes, the probability of an event E is the number of favorable outcomes m divided by the total outcomes n.
- Axiomatic Definition of Probability
A mathematically rigorous definition introduced by Kolmogorov, which is based on three axioms and can handle infinite sample spaces.
- Sample Space (S)
The set of all possible outcomes of a probabilistic experiment.
- Event (F)
A collection of subsets of the sample space, including the sample space itself and the empty set.
- Probability Function (P)
A function that assigns a probability to each event in F.
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