Conditional Probability Step 4: Compute Event Probabilities Pr(A B Pr(A∩ Pr(B) 1 The first equation uses the Product Rule. On the second line, we use the fact that the probability of an event is the sum of the probabilities of the outcomes it contains final line is simplification. The probability that the fair coin was chosen, given that result of the flip was heads, is 1 3 2.2 A Variant of the two coins problem Let's consider a variant of the two coins problem. Someone hands you either a fair coin or a trick coin with heads on both sides. You flip the coin 100 times and see heads every time What can you say about the probability that you flipped the fair coin? Remarkably In order to make sense out of this outrageous claim, let's formalize the problem. The sample space is worked out in the tree diagram below. We do not know the probability that you were handed the fair coin initially- you were just given one coin or the other- o lets call that p result of event A: event B: 100 flips coin gIven given fair flipp to you coin? all heads? probab all heads X p/2 fair coin 1-12 ome tails 1-p trick coin 112 all heads Let a be the event that you were handed the fair coin, and let b be the event that you lipped 100 heads. Now, were looking for Pr(A I B), the probability that you were8 Conditional Probability Step 4: Compute Event Probabilities Pr(A | B) = Pr(A ∩ B) Pr(B) 1 4 = 1 1 4 + 2 1 = 3 The first equation uses the Product Rule. On the second line, we use the fact that the probability of an event is the sum of the probabilities of the outcomes it contains. The final line is simplification. The probability that the fair coin was chosen, given that the result of the flip was heads, is 1/3. 2.2 A Variant of the Two Coins Problem Let’s consider a variant of the two coins problem. Someone hands you either a fair coin or a trick coin with heads on both sides. You flip the coin 100 times and see heads every time. What can you say about the probability that you flipped the fair coin? Remarkably— nothing! In order to make sense out of this outrageous claim, let’s formalize the problem. The sample space is worked out in the tree diagram below. We do not know the probability that you were handed the fair coin initially— you were just given one coin or the other— so let’s call that p. result of 100 flips 1/2100 1/2100 100 1−1/2 event A: given fair coin? event B: flipped all heads? coin given to you fair coin trick coin all heads some tails all heads probability X X X X 1−p p / 2 p − p / 2 100 100 p 1−p Let A be the event that you were handed the fair coin, and let B be the event that you flipped 100 heads. Now, we’re looking for Pr(A | B), the probability that you were