02_probability.pdf - 2 Probability and conditional...

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2Probability and conditional probabilityProbabilityis a rich language for communicating about uncer-tainty. And while most of us have an intuitive notion of what itmeans, it pays to be a bit more specific.A probability is just a number that measures how likely it isthat some event, like rain, will occur. IfAis an event,P(A)is itsprobability:P(coin lands heads) =0.5,P(rainy day in Ireland) =0.85,P(cold day in Hell) =0.0000001, and so forth.Probability from frequencies.Some probabilities are derived fromdata, like the knowledge that a coin comes up heads about50%of the time in the long run, or that11people out of100,000diein a car accident. But it’s also perfectly normal for a probabilityto reflect your subjective assessment or belief about something.Here, you should imagine a stock-market investor who has todecide whether to buy a stock or sell it. The performance of astock over the coming months and years involves a bunch of one-off events that have never happened before, and will never berepeated. But that’s OK. We can still talk about a probability likeP(Apple stock goes up next month). We just have to recognizethat this probability reflects someone’s subjective judgment, ratherthan a long-run frequency from some hypothetical coin-flippingexperiment.Probability from judgment and/or betting markets.If you don’t haveany data, a great way to estimate the probability of some event isto get people to make bets on it. Let’s take the example of the2014mens’ final at Wimbledon, between Novak Djokovic and RogerFederer. This was one of the most anticipated tennis matches inyears. Djokovic, at27years old, was the top-ranked player in theworld and at the pinnacle of the sport. And Federer was—well,Federer! Even at32years old and a bit past his prime, he wasranked #3in the world, and had been in vintage form leading up
10data scienceto the final.How could you synthesize all this information to estimate aprobability likeP(Federer wins)? Well, if you walked into anybetting shop in Britain just before the match started, you wouldbeen quoted odds of20/13on a Federer victory.1To interpret1There are approximately9,000bettingshopsin the United Kingdom. In fact,it is estimated that approximately4%of all retail storefronts in England arebetting shops.odds in sports betting, think “losses over wins.” That is, if Federerand Djokovic played33matches, Federer would be expected towin13of them and lose20, meaning thatP(Federer wins match) =1313+200.4 .The markets had synthesized all the available information for you,and concluded that the pre-match probability of a Federer victorywas just shy of40%. (Djokovic ended up winning in five sets.)Conditional probabilityAnother very important concept is that of aconditional probability.

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