Recitation 18 Problem 2. a Barglesnort makes its lair in one of three caves The Barglesnort inhabits cave 1 with probability ,, cave 2 with probability 1, and cave 3 with probability. a rabbit subsequently moves into one of the two unoccupied caves selected with equal probability. With probability 3, the rabbit leaves tracks at the entranc to its cave. (Barglesnorts are much too clever to leave tracks )What is the probability that the Barglesnort lives in cave 3, given that there are no tracks in front of cave 2? Use a tree diagram and the four-step method Solution. A tree diagram is given below. Let B be the event that the Barglesnort inhabits cave 3, and let T2 be the event that there are tracks in front of cave 2. Taking data from the tree diagram, we can compute the desired probability as follows: Pt(B1|72)=P(B3n 5 In the denominator, we apply the formula Pr(T2)=1-Pr(T2)for convenience Rabbits prob. Bargle's yes-1/2� � � � � � Recitation 18 3 Problem 2. A Barglesnort makes its lair in one of three caves: 1 2 3 The Barglesnort inhabits cave 1 with probability 1 2 , cave 2 with probability 1 , and cave 3 4 with probability 1 . A rabbit subsequently moves into one of the two unoccupied caves, 4 selected with equal probability. With probability 1 , the rabbit leaves tracks at the entrance 3 to its cave. (Barglesnorts are much too clever to leave tracks.) What is the probability that the Barglesnort lives in cave 3, given that there are no tracks in front of cave 2? Use a tree diagram and the four­step method. Solution. A tree diagram is given below. Let B3 be the event that the Barglesnort inhabits cave 3, and let T2 be the event that there are tracks in front of cave 2. Taking data from the tree diagram, we can compute the desired probability as follows: Pr B3 | T2 = Pr B � 3 ∩ � T2 Pr T2 1 1 1 24 12 12 + + = 1 1 1 − 12 − 24 5 = 21 In the denominator, we apply the formula Pr T2 = 1 − Pr (T2) for convenience. 1 2 3 1 3 3 2 1 2 Bargle’s lair Rabbit’s hole rabbit tracks? no yes 1/2 1/2 1/4 1/4 1/2 1/2 1/2 1/2 1/2 1/3 1/3 1/3 1/3 1/3 1/3 2/3 2/3 2/3 2/3 2/3 2/3 prob. 1/12 1/12 1/6 1/6 1/24 1/12 1/24 1/12 1/24 1/12 1/24 1/12 Bargle yes yes yes yes in 3? Tracks at 2? yes yes