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Math Help - binomial distribution

  1. #1
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    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
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  2. #2
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    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}


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