SOLUTION: A bag has 3 red and k white marbles, where k is an (unknown) positive integer. Two of the marbles are chosen at random from the bag. If the probability that the two marbles are the

Algebra ->  Permutations -> SOLUTION: A bag has 3 red and k white marbles, where k is an (unknown) positive integer. Two of the marbles are chosen at random from the bag. If the probability that the two marbles are the      Log On


   

Question 394011: A bag has 3 red and k white marbles, where k is an (unknown) positive integer. Two of the marbles are chosen at random from the bag. If the probability that the two marbles are the same color is 1/2, find k.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
We assume that k%3E=+2, or else the event of getting 2 white marbles is excluded.
The probability of getting 2 red marbles = 3C2%2F%28k%2B3%29C2
The probability of getting 2 white marbles = kC2%2F%28k%2B3%29C2
Hence
3C2%2F%28k%2B3%29C2+%2B+kC2%2F%28k%2B3%29C2+=+1%2F2
<==> 6%2F%28%28k%2B3%29%28k%2B2%29%29+%2B+%28k%28k-1%29%29%2F%28%28k%2B3%29%28k%2B2%29%29+=+1%2F2 after simplifying.
<==> %28k%5E2+-+k+%2B+6%29%2F%28k%5E2+%2B+5k+%2B+6%29+=+1%2F2
<==> k%5E2+-+7k+%2B+6+=+0, after cross-multiplying and simplifying.
<==> (k-1)(k - 6) = 0.
==> k = 1, 6.
Therefore k = 6, the number of white marbles.