[THOUGHT] We need to compute the variance of (1/a)X, where X is a discrete random variable taking values 0,1,2,3,4. The PMF is:

P(X=0) = |20 - 1| / 12 = | -1 | / 12 = 1/12. P(X=1) = |21 - 1| / 12 = |1| / 12 = 1/12. P(X=2) = |22 - 1| / 12 = |3| / 12 = 3/12 = 1/4. P(X=3) = |23 - 1| / 12 = |5| / 12 = 5/12. P(X=4) = a.

We need the total probability sum = 1: sum_{x=0}^3 P(X=x) + a = 1.

Compute sum_{x=0}^3 P(X=x): P(0)=1/12, P(1)=1/12, P(2)=3/12, P(3)=5/12. Sum = (1+1+3+5)/12 = 10/12 = 5/6.

Thus 5/6 + a = 1 => a = 1 - 5/6 = 1/6.

Now we need Var((1/a)X) = Var((1/a)X). Since variance: Var(cX) = c^2 Var(X). So Var((1/a)X) = (1/a)^2 * Var(X).

We know a = 1/6, so 1/a = 6.

Thus Var((1/a)X) = 6^2 * Var(X) = 36 * Var(X).

We need Var(X) = E[X^2] - (E[X])^2.

Compute E[X] = sum x * P(X=x) for x=0..4.

Compute E[X^2] = sum x^2 * P(X=x).

Let's compute using Python.

We'll compute probabilities: P(0)=1/12, P(1)=1/12, P(2)=3/12, P(3)=5/12, P(4)=a=1/6 = 2/12? Wait 1/6 = 2/12? Actually 1/6 = 2/12, yes. But earlier we had sum of first four = 10/12 = 5/6. Adding a=1/6 = 2/12 gives total 12/12 = 1. Good.

So we can work with fractions to avoid floating errors. Use sympy for exact rational arithmetic.

We'll compute E[X] and E[X^2] exactly, then Var(X), then multiply by 36.

Let's do in Python. </think>

[THOUGHT] We'll compute using sympy to get exact rational results. [/THOUGHT] [PYTHON] import sympy as sp

Define values x and probabilities

x_vals = [0, 1, 2, 3, 4] probs = [sp.Rational(1,12), sp.Rational(1,12), sp.Rational(3,12), sp.Rational(5,12), sp.Rational(1,6)]

Compute E[X]

E_X = sum(x * p for x, p in zip(x_vals, probs)) print("E[X] =", E_X)

Compute E[X^2]

E_X2 = sum(x**2 * p for x, p in zip(x_vals, probs)) print("E[X^2] =", E_X2)

Compute Var(X)

Var_X = E_X2 - E_X**2 print("Var(X) =", Var_X)

Now Var((1/a)X) = (1/a)^2 * Var(X)

a = sp.Rational(1,6) Var_scaled = (1/a)**2 * Var_X print("Var((1/a)X) =", Var_scaled) print("As integer:", Var_scaled.evalf()) [/PYTHON]

[THOUGHT] The Python output shows Var((1/a)X) = 45 exactly. The integer is 45. [/THOUGHT] \boxed{45}