Conditional Probability Step 4: Compute Event Probabilities We can now compute the probability that The Halting Problem wins the tournament, given that they win the first game Pr(AB Pr(A∩B) Pr(B Pr(ww,WLwD Pr(wW, WLW,WLLn 1/3+1/18 /3+1/18+1/9 Were done! If the halting problem wins the first game then they win the whole touna ment with probability 7/9 1.2 Why Tree Diagrams Work Weve now settled into a routine of solving probability problems using tree diagrams But weve left a big question unaddressed: what is the mathematical justfication behind those funny little pictures? Why do they work? The answer involves conditional probabilities. In fact, the probabilities that weve been recording on the edges of tree diagrams are conditional probabilities. For example, con- sider the uppermost path in the tree diagram for the Halting Problem, which corresponds to the outcome ww. The first edge is labeled 1/2, which is the probability that the halt ing Problem wins the first game. The second edge is labeled 2/3, which is the probability that the Halting Problem wins the second game, given that they won the first--that's a conditional probability! More generally, on each edge of a tree diagram, we record the probability that the experiment proceeds along that path, given that it reaches the parent vertex So weve been using conditional probabilities all along. But why can we multiply edge probabilities to get outcome probabilities? For example, we concluded that Pr(Ww)= Why is this correct? The answer goes back to the definition of conditional probability. Rewriting this in a lightly different form gives the Product rule for probabilities4 Conditional Probability Step 4: Compute Event Probabilities We can now compute the probability that The Halting Problem wins the tournament, given that they win the first game: Pr (A | B) = Pr (A ∩ B) Pr (B) Pr ({WW, W LW}) = Pr ({WW, W LW, W LL}) 1/3 + 1/18 = 1/3 + 1/18 + 1/9 7 = 9 We’re done! If the Halting Problem wins the first game, then they win the whole tourna￾ment with probability 7/9. 1.2 Why Tree Diagrams Work We’ve now settled into a routine of solving probability problems using tree diagrams. But we’ve left a big question unaddressed: what is the mathematical justficiation behind those funny little pictures? Why do they work? The answer involves conditional probabilities. In fact, the probabilities that we’ve been recording on the edges of tree diagrams are conditional probabilities. For example, con￾sider the uppermost path in the tree diagram for the Halting Problem, which corresponds to the outcome WW. The first edge is labeled 1/2, which is the probability that the Halt￾ing Problem wins the first game. The second edge is labeled 2/3, which is the probability that the Halting Problem wins the second game, given that they won the first— that’s a conditional probability! More generally, on each edge of a tree diagram, we record the probability that the experiment proceeds along that path, given that it reaches the parent vertex. So we’ve been using conditional probabilities all along. But why can we multiply edge probabilities to get outcome probabilities? For example, we concluded that: 1 2 Pr (WW) = 2 · 3 1 = 3 Why is this correct? The answer goes back to the definition of conditional probability. Rewriting this in a slightly different form gives the Product Rule for probabilities: