Probability Review Sep 24, 2014 Interac9on of Events Disjoint HHH, HHT, HTH, THH"
At least two heads TTT, TTH, THT, HTT"
At least two tails Interac9on of Events Not Disjoint HHH, HHT, HTH"
TTT, TTH, THT"
At least two First toss is THH"
heads a tail Beyond Fair Coins Biased coin: P(H): 0.6, P(T): 0.4. In a three-­‐coin toss, what is the probability of 1.  Three heads? 0.63 = 0.216 2.  At least one tail? 1-­‐0.216 = 0.784 3.  At least two heads? 0.6*0.6*0.4*3+0.6^3 = 0.648 4.  No heads? 0.4^3 = 0.064 5.  The first toss being a tail? 0.4 6.  At least two heads or no heads? 0.648+0.064 = 0.712 7.  At least two heads or the first toss is a tail? 0.648+0.4-­‐0.144=0.904 Beyond Fair Coins Biased coin: P(H): 0.6, P(T): 0.4. In a three-­‐coin toss, what is the probability of 1.  The first toss is a tail and there are at least two heads? 0.4*0.6*0.6=0.144 2.  The first toss is a tail given that at least two heads came up? 0.144/0.648 3.  Ge^ng at least two heads given the first toss is a tail? ? 0.144/0.4 = 0.36 P(capitalized and noun) = 0.03 P(capitalized) = 0.05 P(noun) = 0.30 •  Probability of a non-­‐capitalized noun? 0.27 •  Prob. of a capitalized word that is not a noun? 0.02 •  Prob. of a word that’s capitalized or a noun? 0.32 •  Prob. of a word that’s neither cap. nor a noun? 0.68 •  Prob. that a word is cap. given that it’s a noun? 0.1 •  Prob. that is a word is a noun given that it’s cap.? 0.6 Part of Speech Example •  Prob. of seeing a noun = 0.5 •  Prob. of seeing a verb = 0.1 •  Prob. of seeing run given we see a noun = 0.03 •  Prob. of seeing run given we see a verb = 0.08 •  Overall probability of seeing run? 0.023 •  Prob. of seeing a noun given that you saw run? 0.5*0.03/0.023 = 0.652 Bayes Rule •  Most individuals don’t hunt dwarves P(hunting) = 0.001"
•  When a dwarf is hunted, there is a 70% chance that it was hunted by a goblin •  Goblins comprise 25% of the popula9on •  Therefore, goblins tend to be dwarf hunters (??) P(hun9ng|goblin) = P(goblin|hun9ng)P(hun9ng)/P(goblin) = 0.7*0.001/.25 = 0.0028 Bayes Rule Posterior probability P(hun9ng|goblin) = P(goblin|hun9ng)P(hun9ng) P(goblin) Likelihood of evidence Prior probability Marginal probability of evidence Expecta9on •  Defined for outcomes with a numerical value •  E[X] = P(outcome A of X)*Value(A) + P(outcome B of X)*Value(B)+… •  Linearity of Expecta9on –  E[cX] = cE[X] –  E[X+Y] = E[X]+E[Y] What is the expected sum of numbers on 10 dice? 35 What is the expected sum of numbers when you roll a 6-­‐sided and an 8-­‐sided die together? 8