SOLUTION: Question is: On National Public Radio, the Weekend Edition program on Sunday, September 7, 1991, posed the following probability problem: Given a certain number of balls, of wh

Algebra ->  Probability-and-statistics -> SOLUTION: Question is: On National Public Radio, the Weekend Edition program on Sunday, September 7, 1991, posed the following probability problem: Given a certain number of balls, of wh      Log On


   

Question 950234: Question is:
On National Public Radio, the Weekend Edition program on Sunday, September 7, 1991, posed the following probability problem: Given a certain number of balls, of which some are blue, pick 5 at random. The probability that all 5 are blue is 1/2. Determine the original number of balls and decide how many were blue.
Please show work. Thank you.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's call the number of blue balls X and the number of black balls Y.
The probability of choosing the first blue ball would then be,
P%28B1%29=X%2F%28X%2BY%29
Similarly for the second through the fifth,
P%28B2%29=%28X-1%29%2F%28X%2BY-1%29
P%28B3%29=%28X-2%29%2F%28X%2BY-2%29
P%28B4%29=%28X-3%29%2F%28X%2BY-3%29
P%28B5%29=%28X-4%29%2F%28X%2BY-4%29
So then the probability of getting all 5 is,
P=P%28B1%29%2AP%28B2%29%2AP%28B3%29%2AP%28B4%29%2AP%28B5%29

That seems pretty daunting, let's take a guess and you'll see that it's not really.
Let's assume Y=1, then the equation becomes,

%28X-4%29%2F%28X%2B1%29=1%2F2
2%28X-4%29=X%2B1
2X-8=X%2B1
X=9
Is that the only answer?
How about Y=2, then the equation becomes,

Now it becomes more daunting.
%28%28X-3%29%28X-4%29%29%2F%28%28X%2B2%29%28X%2B1%29%29=1%2F2
2%28X%5E2-7X%2B12%29=X%5E2%2B3x%2B2
X%5E2-17X-22=0
This equation doesn't have an integer solution.
So there's one solution but I'm not sure it's unique.