In an urn, there are several colored balls, with equal numbers of each color. We add 14 balls which are all of the same new color, that is different from those in the urn. It is calculated that the probability of drawing, without replacement, two balls of the same color is the same (when compared before and after the balls are added). How many balls are there in the urn initially?
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To solve this problem, we'll break it down into smaller steps.
Let's assume there are "x" balls of each color initially in the urn, and the number of colors is represented by "n". So, the total number of balls in the urn initially would be "n * x".
After adding 14 balls of a new color, the total number of balls in the urn becomes "n * (x + 1) + 14".
Now, let's calculate the probability of drawing two balls of the same color before and after the 14 balls are added.
Before adding the new balls, the probability of drawing two balls of the same color without replacement can be calculated as follows:
The probability of drawing the first ball of any color is "x / (n * x)" since there are "x" balls of each color initially, and the total number of balls is "n * x".
The probability of drawing the second ball of the same color without replacement is "(x - 1) / (n * x - 1)" since there is one less ball of that color in the urn.
So, the probability before adding the 14 balls is: "x / (n * x) * (x - 1) / (n * x - 1)".
After adding the 14 balls, the probability of drawing two balls of the same color without replacement is still the same. Therefore, the probability after adding the 14 balls can be calculated as follows:
The probability of drawing the first ball of any color is now "x / (n * (x + 1) + 14)" since the total number of balls has increased.
The probability of drawing the second ball of the same color without replacement is "(x - 1) / (n * (x + 1) + 14 - 1)".
So, the probability after adding the 14 balls is: "x / (n * (x + 1) + 14) * (x - 1) / (n * (x + 1) + 14 - 1)".
Since the probability before and after adding the 14 balls is the same, we can set the two expressions equal to each other:
"x / (n * x) * (x - 1) / (n * x - 1) = x / (n * (x + 1) + 14) * (x - 1) / (n * (x + 1) + 14 - 1)"
Now we can solve this equation to find the value of "x" (the number of balls of each color initially).
This equation may seem complicated to solve algebraically, but we can use numerical methods or solve iteratively to find the value of "x".