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Video transcript

consider the following story Bob is in a room and he has two coins one fair coin and one double sided coin he picks one at random flips it and shouts the result hey now what is the probability that he flipped the fair coin to answer this question we need only rewind and grow a tree the first event he picks one of two coins so our tree grows two branches leading to two equally likely outcomes fair or unfair the next event he flips the coin we grow again if he had the fair coin we know this flip can result in two equally likely outcomes heads and tails while the unfair coin results in two outcomes both heads our tree is finished and we see it has four leaves representing four equally likely outcomes the final step new evidence he says hey whenever we gain evidence we must trim our tree we cut any branch leading to tails because we know tails did not occur and that is it so the probability that he chose the fair coin is the one fair outcome leaving two heads divided by the three possible outcomes leading to heads or one-third what happens if he flips again and reports Hey remember after each event our tree grows the fair coin leaves result in two equally likely outcomes heads and tails the unfair coin leaves result in two equally likely outcomes heads and heads after we hear the second hey we cut any branches leading to tails therefore the probability the coin is fair after two heads in a row is the one fair oh come leading to heads divided by all possible outcome leading to heads or one fifth notice our confidence in the fair coin is dropping Moorhead's occur though realize it will never reach zero no matter how many flips occur we can never be 100% certain the coin is unfair in fact all conditional probability questions can be solved by growing trees let's do one more to be sure bob has three coins two are fair one is biased which is weighted to land heads two thirds of the time and tails one-third he chooses a coin at random and flips it hey now what is the probability he chose the biased coin let's rewind and build a tree the first event choosing the coin can lead to three equally likely outcomes fair coin fair coin and unfair coin the next event the coin is flipped each fair coin leads to two equally likely leaves heads and tails the biased coin leads to three equally likely leaves two representing heads and one representing tails now the trick is to always make sure our tree is balanced meaning an equal amount of leaves growing out of each branch to do this we simply scale up the number of branches to the least common multiple for two and three this is six and finally we label our leaves the fair coin now splits into six equally likely leaves three heads and three tails for the biased coin we now have two tail leaves and four head leaves and that is it when Bob shouts the result hey this new evidence allows us to trim all branches leading to tails since tails did not occur so the probability that he chose the biased coin given heads occur well four leaves can come from the bias coin divided by all possible leaves 4/10 or 40% when in doubt it's always possible to answer conditional probability questions by Bayes theorem it tells us the probability of event a given some new evidence B though if you forgot it no worries you need only know how to grow stories with trimmed trees