1. binomial distribution

Suppose that each day the price of a stock moves up one dollar with probability 1/3 and moves down one dollar with probability 2/3. If the price fluctuation from one day to another are independent, what is the probability that after six days the stock has its original price? Thanks

2. Hello, jojo_jojo!

Suppose that each day the price of a stock moves up $1 with probability 1/3 and moves down$1 with probability 2/3.
If the price fluctuations from one day to another are independent,
what is the probability that after six days the stock has its original price?

We are given: . $P(\text{Up}) = \tfrac{1}{3},\;\;P(\text{Down}) = \tfrac{2}{3}$

If the stock is back to original price,
. . there were three Ups and three Downs, in some order.

There are: . ${6\choose3,3} = 20$ possible orders.
The probability of 3 Ups and 3 Downs is: . $\left(\tfrac{1}{3}\right)^3\left(\tfrac{2}{3}\righ t)^3 \:=\:\frac{8}{729}$

Therefore: . $P\text{(3 Ups, 3 Down, any order)} \;=\;20\cdot\frac{8}{729} \;=\;\frac{160}{729}$