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Probability in Daily Life: Understanding Randomness and Independence

Explore the concept of probability in our daily lives, including the idea of randomness and independence in events. Learn how to calculate probabilities using logic and sample trials. Discover how the law of large numbers makes outcomes more predictable with a large number of trials.

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Probability in Daily Life: Understanding Randomness and Independence

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  1. Chapter Five 5.1 Probability in our Daily Lives

  2. What is Random? • Which of these list is a “random” list of results when flipping a fair coin 10 times? • A) T H T H T H T H T H • B) T T H T H T T H H T • C) H HHHHHHHH • Answer: D) All of the above!

  3. Probability: • With a small number of trials the outcome of random events can vary widely from what we expect them to be. • Examples: • Coin Flips • Winning Streaks • Others

  4. Probability: • However with a Large number of event, Called Trials, the proportion of times the event occurs will become more predictable. • Examples: • Casino games • Others

  5. Law of Large Numbers: • With a randomized experiment, the probability of a particular outcome is the proportion of times that outcome would occur in a long run of observations. • Outcome: The result of a specific trial. • Long run: Very large, often theoretically large, number of trials.

  6. Independence: • We are going to try a few coin flips: • Flip One: Tails • Flip Two: Heads • Flip Three: Heads • Flip Four: Heads • Flip Five: Heads • Flip six: Heads • No matter what happened before the odds will always be 50::50.

  7. Independent Trials: • Different trials of a random phenomenon are independent if the outcome of any one trial is not effected by the outcome of any other trial. Independent Examples: Dependent Examples:

  8. Finding Probabilities: • 1) Logic, symmetry, structure. • Best for simple well understood situations. • 2) Sample trials. • Good for complex situations we can model.

  9. Next Time Finding Probabilities. “What is the chance of rolling at least one 6 on six roles?”

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