Exam-Style Question on Probability

Some orange fish and green fish are swimming in a large tank in the enterance to a restaurant. The number of orange fish is a single digit number as is the number of green fish.

The chef occasionally takes a fish from the tank at random. So far today he has taken out two fish.

The tank initially contains r orange fish and g green fish.

Let \( P(GG) \) represent the probability of drawing two green fish from the tank without replacement.

It is known that \( P(GG) = \frac{1}{5} \).

(a) Show that \( 4g^2 - (4 + 2r)g + r - r^2 = 0 \).

(b) By solving the equation in part (a), show that \( g = \dfrac{(2+r) \pm \sqrt{5r^2 + 4}}{4} \).

(c) Find two pairs of values for r and g that satisfy the condition \( P(GG) = \frac{1}{5} \).

On a different day the chef randomly takes three fish out of the tank. The tank initially contained 10 orange fish and g green fish.

Let \( P(GGG) \) represent the probability of taking three green fish from the tank without replacement.

(d) Find an expression for \( P(GGG) \) in terms of g.

A green fish is added so that the tank now contains 10 orange fish and \( g + 1 \) green fish. The probability of taking three green fish from the tank without replacement is now twice the probability expressed in part (d).

(e) Find the initial number of green fish in the tank on this particular day.