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Given probability, solve unknown number of counters

  1. Jun 9, 2017 #1

    CWatters

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    Not my homework exactly but it's homework like so..

    1. The problem statement, all variables and given/known data

    There are some red and white counters in a bag. At the start there are 7 red and the rest white. Alfie takes two counters at random without putting any back. The probability that the first is white and the second red is 21/80.

    How many white counters were in the bag at the start?

    2. Relevant equations

    P(W&R) = P(W) * P(R)

    3. The attempt at a solution

    I have the solution but it took well over the 3.5 mins budgeted for each question in the paper. Along the way I had to use the general equation for solving a quadratic. Did I miss obvious factors of Eqn (3)?

    Let: T = Total number of counters, W = number of white counters

    T = 7 + W .........................................................(1)
    P(W&R) = (W/T) * (7/(T-1)) = 21/80...................(2)

    Two equations and 2 unknowns so should be solvable..

    Sub for T in (2)...

    (W/(7+W)) * (7/(7+W-1) = 21/80
    Multiply out..
    7W / (W2 + 13W + 42) = 21/80
    7*80W = 21*(W2 + 13W + 42)
    80W = 3*(W2 + 13W + 42)
    80W - 3*(W2 + 13W + 42) = 0
    80W - 3W2 - 39W -126 = 0
    finally the quadratic..
    3W2 - 41W + 126 = 0......................(3)

    Then using -b+/-Sqrt(b2 -4ac)/2a

    I got answers 14/3 and 9. Answer is 9 because 14/3 isn't a whole number. I checked it's correct by putting W=9 and T=16 back into (2).
     
  2. jcsd
  3. Jun 9, 2017 #2

    PeroK

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    I don't immediately see a shortcut.
     
  4. Jun 9, 2017 #3

    haruspex

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    Not sure that it is any quicker, but having got to (w+7)(w+6)=80w/3 it is clear that 3|w, and then immediately that 9|w, so can rewrite as (9w'+7)(3w'+2)=80w'. Obviously w'=0 is too small and 1 is too much, so....
     
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