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Introduction to Probability: Understanding Chance and Likelihood

Learn the classical definition of probability, how to communicate and reason about probabilities, and interpret relative frequencies. Explore frequency trees and solve probability scenarios using real-life examples. Understand how to calculate probabilities given specific conditions.

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Introduction to Probability: Understanding Chance and Likelihood

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  1. Introduction to Probability

  2. What is probability? • Classical definition: • the ratio of “favorable” to equally probable cases. • “favorable”:the kind you’re interested in. • Probability of getting heads on flipping a fair coin: 1/2 (heads is 1 of 2 possibilities)

  3. What is the probability of getting heads twice on two tosses? Fourpossibilities: • one out of four (1/4): just Case 1 • What is the probability of getting heads at least once on two tosses? • 3/4: Cases 1, 2, and 3

  4. Communicating and reasoning about probabilities • 25% • 25 out of 100 • 1/4 • .25

  5. Communicating and reasoning about probabilities • “If you take Prozac you have a 30–50% chance of negative sexual side effects.” • “Out of every 10 people who take Prozac, 3 to 5 of them develop negative sexual side effects.” • Studies show that relative frequencies are easier to think about.

  6. So translate probabilities into relative frequencies! • Bonus benefit: will make you get explicit about the reference class.

  7. “There’s a 30% chance of rain today.” • “There’s an 80% chance you’ll survive this experimental surgery that’s never been done before.” • “30% of days like today rain.” • “I’m 80% sure.” • not a probability claim at all, but statement of confidence

  8. Frequency trees • Suppose there is a 40% chance it will rain today and a 90% chance you’ll get wet if it rains. What is the probability that you get wet today? • Solve with a frequency tree.

  9. Start with a nice round number. Translate 40% chance of rain into frequencies. 100 days 90% chance you get wet if it rains. Suppose: 20% chance you get wet if it doesn’t rain. 40 rains 60 doesn’t rain But that’s not the only way you might get wet—you might get sprayed by a hose, etc. even if it doesn’t rain. 36 get wet 4 stay dry 12 get wet 48 stay dry Of these 100 days, you get wet on 48 (36+12) of them. There’s a 48% chance you’ll get wet.

  10. A man between 35 and 44 years old has a 0.6% probability of having prostate cancer. If he has it, there is a 58% chance that the PSA test will catch it. If he doesn’t have it, there is a 23.5% chance that he will test positive nonetheless. X (a 42-year-old man) receives a positive PSA test. What is the probability that he has prostate cancer?

  11. base rate = .6% (how many have it regardless of test) Probability of having cancer given positive screen is 35/2371. 10,000 men sensitivity of test = 58% 60 have cancer 9,940 don’t approx. 1.5%! false positives = 23.5% 35 test + 25 test - 2336 test + 7604 test - 2371 men test positive (2336+35) of these, 35 actually have it.

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