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%**

**_____________________________________________**

**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%**

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