Conditional Probability always comes up heads. Now you choose a coin at random so that you're equally likely to pick each one. If you flip the coin you select and get heads, then what is the probability that you flipped the fair coin? This is another a posteriori problem since we want the probability of an event(that the fair coin was chosen) given the outcome of a later event(that heads came up). Intuition may fail us, but the standard four-step method works perfectly well Step 1: Find the Sample Space The sample space is worked out in the tree diagram below l/4 1/2 1/2 fair 1/2 unfa 1/2 event a: event b: ever outcome choice of result probability fair coin? heads? coin flip Step 2: Define Events of Interest Let a be the event that the fair coin was chosen. Let b the event that the result of the flip was heads. The outcomes in each event are marked in the figure. We want to compute Pr(AI B), the probability that the fair coin was chosen, given that the result of the flip was head Step 2: Compute Outcome Probabilities First, we assign probabilities to edges in the tree diagram. Each coin is chosen with prob ability 1/2. If we choose the fair coin, then head and tails each come up with probability 1/2. If we choose the trick coin, then heads comes up with probability 1. By the Product Rule, the probability of an outcome is the product of the probabilities on the correspond ing root-to-leaf path. All of these probabilities are shown in the tree diagramConditional Probability 7 always comes up heads. Now you choose a coin at random so that you’re equally likely to pick each one. If you flip the coin you select and get heads, then what is the probability that you flipped the fair coin? This is another a posteriori problem since we want the probability of an event (that the fair coin was chosen) given the outcome of a later event (that heads came up). Intuition may fail us, but the standard four­step method works perfectly well. Step 1: Find the Sample Space The sample space is worked out in the tree diagram below. fair unfair choice of coin result H flip 1/2 1/2 1/2 1/2 H T 1/2 1/4 1/4 event A: outcome probability fair coin? event B: outcome heads? event chose A B? Step 2: Define Events of Interest Let A be the event that the fair coin was chosen. Let B the event that the result of the flip was heads. The outcomes in each event are marked in the figure. We want to compute Pr(A | B), the probability that the fair coin was chosen, given that the result of the flip was heads. Step 2: Compute Outcome Probabilities First, we assign probabilities to edges in the tree diagram. Each coin is chosen with prob￾ability 1/2. If we choose the fair coin, then head and tails each come up with probability 1/2. If we choose the trick coin, then heads comes up with probability 1. By the Product Rule, the probability of an outcome is the product of the probabilities on the correspond￾ing root­to­leaf path. All of these probabilities are shown in the tree diagram