Introduction to Probability 2018 Taylor Francis What is
Introduction to Probability © 2018 Taylor & Francis
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) © 2018 Taylor & Francis
• What is the probability of getting heads twice on two tosses? • Four possibilities: Case 1 Case 2 Case 3 Case 4 HH HT TH TT • 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 © 2018 Taylor & Francis
Communicating and reasoning about probabilities • 25% • 25 out of 100 • 1/4 • . 25 © 2018 Taylor & Francis
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. © 2018 Taylor & Francis
• So translate probabilities into relative frequencies! • Bonus benefit: will make you get explicit about the reference class. © 2018 Taylor & Francis
• “There’s a 30% chance of rain today. ” • “ 30% of days like today rain. ” • “There’s an 80% chance you’ll survive this experimental surgery that’s never been done before. ” • “I’m 80% sure. ” • not a probability claim at all, but statement of confidence © 2018 Taylor & Francis
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. © 2018 Taylor & Francis
Start with a nice round number. 90% chance you get wet if it rains. 36 get wet 40 rains 4 stay dry 100 days Translate 40% chance of rain into frequencies. Suppose: 20% chance you But that’s not the only wayif it get wet you might get wet—you doesn’t 12 get sprayed 48 by arain. might hose, get dry etc. wet evenstay if it doesn’t rain. 60 doesn’t rain Of these 100 days, you get wet on 48 (36 + 12) of them. There’s a 48% chance you’ll get wet. © 2018 Taylor & Francis
• 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? © 2018 Taylor & Francis
Probability of having cancer given positive screen is 35/2371. 60 approx. have 1. 5%! cancer 35 test + base rate =. 6% (how many have it regardless of test) 10, 000 men 25 test - 9, 940 don’t 2336 test + sensitivity of test = 58% 7604 test - false positives = 23. 5% 2371 men test positive (2336 + 35) of these, 35 actually have it. © 2018 Taylor & Francis
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