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Questions and Answers
What is a random experiment?
An experiment that can be repeated under the same conditions where the outcome cannot be predicted with certainty.
What is the sample space denoted by in probability?
What are events in the context of probability?
Subsets of the sample space that we are interested in studying.
The probability measure of an event can be equal to or approximate the relative frequency as the number of experiments increases.
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Define relative frequency.
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In Example 1.1.1, what are the possible outcomes when tossing a coin?
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In the cast of one red die and one white die, the sample space consists of ____ ordered pairs.
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What happens to the relative frequency as the number of experiments increases?
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Study Notes
Introduction to Probability
- Probability models are essential for analyzing phenomena through repeated experimentation under consistent conditions.
- Investigations may involve various fields such as medical research, economics, and agronomy, each requiring experimental processes.
Random Experiments and Sample Spaces
- A random experiment is characterized by outcomes that cannot be predicted with certainty before the experiment.
- The experimental space or sample space (denoted as C) comprises all possible outcomes of a random experiment.
- Example: Tossing a coin has a sample space C = {H, T} representing heads (H) and tails (T).
- Example: Casting one red die and one white die yields a sample space of 36 ordered pairs, C = {(1, 1), (1, 2), ..., (6, 6)}.
Events and Their Occurrence
- Elements of the sample space are typically denoted by small Roman letters (e.g., a, b, c).
- Events are subsets of the sample space denoted by capital Roman letters (e.g., A, B, C).
- The occurrence of an event (like rolling a sum of 7 with two dice) indicates that the result falls within the selected subset.
Relative Frequency and Probability
- After conducting N trials of a random experiment, the frequency of an event A is measured by how often it occurs (f).
- The relative frequency of event A is calculated as f/N, representing the proportion of times A occurs within N trials.
- Relative frequencies may fluctuate with small sample sizes but stabilize as N increases.
- A value p can be associated with event A, interpreted as the long-term probability that A occurs in future trials.
- Probability terminology varies but generally refers to the likelihood of event A, resulting from the relative frequency in a considerable number of trials.
Examples of Probability Calculation
- In a scenario where the dice are cast 400 times with a resulting frequency of 60 for event B (sum of 7), the relative frequency is f/N = 60/400 = 0.15.
- This suggests that the probability p associated with event B (the sum being 7) is approximately 0.15.
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