The random variable X represents the number of successes in 10 independent Bernoulli trials - HSC - SSCE Mathematics Extension 1 - Question 6 - 2021 - Paper 1

We start by calculating r, which is the probability of obtaining at least one success in 10 trials. Since p = 0.9, we first need to find the probability of the complementary event, which is obtaining zero successes (X = 0).

The probability of getting zero successes in n trials can be calculated using the formula: $P(X = 0) = (1 - p)^n$

Substituting into this, we have: $P(X = 0) = (1 - 0.9)^{10} = 0.1^{10} = 0.0000000001.$

Now, we find r: $r = P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.0000000001 ≈ 1.$

Thus, since r is extremely close to 1, we conclude that r > 0.9.