The random variable X represents the number of successes in 10 independent Bernoulli trials - HSC - SSCE Mathematics Extension 1 - Question 8 - 2021 - Paper 1

To determine the value of $r = P(X ≥ 1)$, we can first calculate its complement:

$P(X < 1) = P(X = 0) = (1 - p)^{n} = (1 - 0.9)^{10} = 0.1^{10} = 0.0000000001$

Thus, we have:

$P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.0000000001 ≈ 1$

Since $r$ is almost equal to 1, it follows that $r > 0.9$ and specifically is definitely in the range $0.1 < r < 0.9$. Therefore, the correct choice is C.