In: Math
An investment analyst has tracked a certain fund and found that it moves independently day to day, up or down a point. The probability of going up is 75%. What is the probability that four days from now, the price will be the same as now?
The probability og going up = 75% = 3/4 and the probability of going down = 25% = 1 / 4
For the price to be the same as now, It has to have 2 Ups and 2 Downs in the 4 days in different combinations
Therefore the total number of ways = 4! / (2!*2!) = 24 / 4 = 6 ways and each will have a probability of (3/4)2 * (1 /4)2 = 9/256
Therefore the required probability = 6 * (9 / 256) = 54 / 256 = 0.2109 = 21.09%
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The second way to do it is to list all the outcomes as below
D D U U = (1 / 4) * (1 / 4) * (3 / 4) * (3 / 4) = 9 / 256
D U D U = (1 / 4) * (3 / 4) * (1 / 4) * (3 / 4) = 9 / 256
U D D U = (3 / 4) * (1 / 4) * (1 / 4) * (3 / 4) = 9 / 256
U U D D = (3 / 4) * (3 / 4) * (1 / 4) * (1 / 4) = 9 / 256
U D U D = (3 / 4) * (1 / 4) * (3 / 4) * (1 / 4) = 9 / 256
D U U D = (1 / 4) * (3 / 4) * (3 / 4) * (3 / 4) = 9 / 256
Therefore the required probability = 6 * (9 / 256)
= 54 / 256 = 0.2109 = 21.09%