I'll probably never understand probability?
Hey,
I have been doing some probability and we have moved onto conditional probability with mutual exclusivity, independence and such like. Im pretty ok when I have the variables to the left of the = sign. However my reasoning breaks down when i have one variable and an answer but not the 2nd variable.
For e.g.
A bag contains 5 black and x blue marbles. One marble is drawn at random without replacement and then another drawn. If the probability of drawing 2 blue marbles is 1/6, how many blue marbles are in the bag?
The answer is 4 but I dont know a good routine to get to that - can anybody help with a way to achieve this?
Hope this makes sense, let me know if you need more info.
Cheers.
Remember that probability is just the number of times the desired event can happen, divided by the total number of possible events. There are a total of 5+x marbles in the bag. Of these, x are blue. So the probabiltiy of pulling a blue on the first try is x/(5+x).
After taking a first marble, this would leave 5+x - 1 = 4+x marbles in the bag. If you chose a blue the first time, then there would be (x-1) blues left. The probability of pulling out a blue marble now is (x-1)/(4+x).
So to find the probability of pulling a second blue marble, given that you've pulled out a first blue marble, is
[ x/(5+x) ] * [ (x-1)/(4+x) ] = 1/6
Solving this for x gives:
x(x-1) / (5+x)(4+x) = 1/6
6x(x-1) = (5+x)(4+x)
6x^2 - 6x = 20 + 9x + x^2
5x^2 - 15x - 20 = 0
x^2 - 3x - 4 = 0
(x - 4)(x + 1) = 0
x = 4, -1
Since having "-1" marbles doesn't make any sense, the answer is x=4.
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