Lecture 23 - Chapter 19 Uncertainty and Information risk is...

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1 Chapter 19: Uncertainty and Information risk is present when future events occur with measurable probability uncertainty is present when the likelihood of future events is indefinite or incalculable Chapter 19: Uncertainty and information A. Risk and expected value Suppose we flip a coin 10 times Count 3 heads, 7 tails fraction of heads = 0.30 Try a second time Count 6 heads, 4 tails fraction of heads = 0.60 Third trial: fraction of heads = 0.50 Suppose we try 100 coin flips and count the fraction of heads: Trial 1 = 0.48 Trial 2 = 0.54 Trial 3 = 0.56
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2 Fraction of heads as a function of the number of coin flips 0.5001 0.5011 0.5300 0.5700 0.6000 Trial 4 0.4989 0.5079 0.5430 0.5600 0.5000 Trial 3 0.4980 0.5010 0.5230 0.5400 0.6000 Trial 2 0.5039 0.5012 0.4970 0.4800 0.3000 Trial 1 100,000 10,000 1,000 100 10 number of observations 10 1 10 2 10 3 10 4 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of observations Fraction of successes Fraction of successes as a function of number of observations (True probability = 0.5, Trial 1) Fraction of successes as a function of number of observations (True probability = 0.5, Trial 2) 10 1 10 2 10 3 10 4 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of observations Fraction of successes Fraction of successes as a function of number of observations (True probability = 0.5, Trial 3) 10 1 10 2 10 3 10 4 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of observations Fraction of successes Conclusions: When I flip a coin a finite number of n times, I don’t know for sure the fraction of heads • If n is very big, the fraction is going to be very close to 1/2 What’s the fraction of successes look like if true probability of success is 0.8?
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3 Fraction of successes as a function of number of observations (True probability = 0.8, Trial 1) 10 1 10 2 10 3 10 4 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of observations Fraction of successes Suppose I could take the following gamble: with probability 0.8, I receive $2 with probability 0.2, I receive nothing
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