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 $r = P(X \geq 1)$, we start by calculating the probability of getting no successes at all (i.e., $X = 0$) in 10 trials. The probability of failure in each trial is $1 - p = 0.1$. Therefore, the probability of getting 0 successes is:

$P(X = 0) = (0.1)^{10} = 0.0000000001$

Now, the probability of getting at least one success is:

$P(X \geq 1) = 1 - P(X = 0) = 1 - (0.1)^{10} \approx 1$

Given this calculation, we conclude that the best description of $r$ is in the range $r > 0.9$ but very close to 1, which corresponds to option C: $0.1 < r < 0.9$.