Answer: There are 4 male goats.
Step-by-step explanation:
We know that n of the 10 goats are male.
The probability that in a random selection, the selected goat is a male, is equal to the quotient between the number of male goats (n) and the total number of goats (10)
The probability is;
p = n/10
Now the total number of goats is 9, and the number of male goats is n -1
then the probability of selecting a male goat again is:
q = (n-1)/9
The joint probability (the probability that the two selected goats are male) is equal to the product of the individual probabilities, this is
P = p*q = (n/10)*((n-1)/9)
And we know that this probability is equal to 2/15
Then we have:
(n/10)*((n-1)/9) = 2/15
(n*(n-1))/90 = 2/15
n*(n-1) = 90*2/15 = 12
n^2 - n = 12
n^2 - n - 12 = 0
This is a quadratic equation, we can find the solutions if we use Bhaskara's formula:
For an equation:
a*x^2 + b*x + c = 0
The two solutions are given by:
![x = \frac{-b +- \sqrt{b^2 - 4*a*c} }{2*a}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-b%20%2B-%20%5Csqrt%7Bb%5E2%20-%204%2Aa%2Ac%7D%20%7D%7B2%2Aa%7D)
For our case, the solutions will be:
![n = \frac{1 +- \sqrt{(-1)^2 - 4*1*(-12)} }{2*1 } = \frac{1+- 7}{2}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7B1%20%2B-%20%5Csqrt%7B%28-1%29%5E2%20-%204%2A1%2A%28-12%29%7D%20%7D%7B2%2A1%20%7D%20%3D%20%5Cfrac%7B1%2B-%207%7D%7B2%7D)
The two solutions are:
n = (1 - 7)/2 = -3 (this solution does not make sense, we can not have a negative number of goats)
The other solution is:
n = (1 + 7)/2 = 4
This solution does make sense, this means that we have 4 male goats.