A drunkard is standing one step away from the cliff on his left. He moves randomly,...

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A drunkard is standing one step away from the cliff on his left. He moves randomly, one step at a time and independently, either to the right (away from the cliff) with probability p or left (toward the cliff) with probability 1- p. Denote by a the probability that he eventually falls down the cliff (i.e. reaching the point 0). (a) Find a for the special cases of p= 0 and p = 1. cliff→ 1 0 1 2 3 4 (b) In general, show that a satisfies the following equation a=pa² +1-p. (Hint: law of total probability) (c) Solve this quadratic equation for a. Which solution makes sense? You may assume that a depends continuously on p and apply part (a). Plot a as a function of p over [0, 1]. A drunkard is standing one step away from the cliff on his left. He moves randomly, one step at a time and independently, either to the right (away from the cliff) with probability p or left (toward the cliff) with probability 1- p. Denote by a the probability that he eventually falls down the cliff (i.e. reaching the point 0). (a) Find a for the special cases of p= 0 and p = 1. cliff→ 1 0 1 2 3 4 (b) In general, show that a satisfies the following equation a=pa² +1-p. (Hint: law of total probability) (c) Solve this quadratic equation for a. Which solution makes sense? You may assume that a depends continuously on p and apply part (a). Plot a as a function of p over [0, 1]. A drunkard is standing one step away from the cliff on his left. He moves randomly, one step at a time and independently, either to the right (away from the cliff) with probability p or left (toward the cliff) with probability 1- p. Denote by a the probability that he eventually falls down the cliff (i.e. reaching the point 0). (a) Find a for the special cases of p= 0 and p = 1. cliff→ 1 0 1 2 3 4 (b) In general, show that a satisfies the following equation a=pa² +1-p. (Hint: law of total probability) (c) Solve this quadratic equation for a. Which solution makes sense? You may assume that a depends continuously on p and apply part (a). Plot a as a function of p over [0, 1]. A drunkard is standing one step away from the cliff on his left. He moves randomly, one step at a time and independently, either to the right (away from the cliff) with probability p or left (toward the cliff) with probability 1- p. Denote by a the probability that he eventually falls down the cliff (i.e. reaching the point 0). (a) Find a for the special cases of p= 0 and p = 1. cliff→ 1 0 1 2 3 4 (b) In general, show that a satisfies the following equation a=pa² +1-p. (Hint: law of total probability) (c) Solve this quadratic equation for a. Which solution makes sense? You may assume that a depends continuously on p and apply part (a). Plot a as a function of p over [0, 1].