Anthony owns Stock A. Suppose that on any given day, the price of Stock A will e

Question

Anthony owns Stock A. Suppose that on any given day, the price of Stock A will either go up or down by 1 unit with equal probability. Also assume that the daily movements in price are independent over different days. What is the probability that the stock price has gone up by 1 unit after 3 days? Suppose that after 3 days, the stock price has gone down by 1 unit. What is the probability that the price went up on the first day? Anthony's friend, Brandon, owns Stock B. On each day, the price of Stock B will either go up or down by 1 unit. The probability that the price will go up on a single day is 0.3. Again, the daily movements in price are independent over different days. One of the two stocks is randomly chosen. Suppose that after 3 days the price of the chosen stock has gone down by 1 unit. What is the probability that the selected stock is Stock A?

Explanation / Answer

a) probabilty that stock price goes up PA(U)=P(price goes up then goes down then goes up+price goes down then goes up then goes up+price goes up then goes up then goes down)=P(udu+duu+uud) =(1/2)*(1/2)*(1/2)+(1/2)*(1/2)*(1/2)+(1/2)*(1/2)*(1/2) =3/8

b)similarly probabilty that price goes down PA(D) =P(price goes down then goes up then goes down+price goes up then goes down then goes down+price goes down then goes down then goes up) =P(dud+udd+ddu) =3/8

hence probabilty that price goes down given price went up on first day =P(udd)/P(dud+udd+ddu) =(1/8)/(3/8) =1/3

c)probabilty of stock B going up PB(U)=P(udu+duu+uud) =0.3*0.7*0.3+0.7*0.3*0.3+0.3*0.3*0.7 =0.189

hence probabilty of a randomely chosen stock going up P(U)=(1/2)*PA(U)+(1/2)*PB(U) =(1/2)*(3/8)+(1/2)*0.189 =0.282

hence probability that selected stock is given stock has gone up =(1/2)*PA(U)/P(U) =0.665

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