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Not my homework exactly but it's homework like so..
1. Homework Statement
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?
P(W&R) = P(W) * P(R)
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).
1. Homework Statement
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?
Homework Equations
P(W&R) = P(W) * P(R)
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).