A standard six-sided die is rolled 12 times - HSC - SSCE Mathematics Extension 1 - Question 2 - 2023 - Paper 1

To find the probability that at least 9 out of 12 rolls of a standard six-sided die result in a 2, we can use the binomial distribution. The probability of rolling a 2 in a single roll is $p = \frac{1}{6}$.

Let $X$ be the random variable denoting the number of times we roll a 2 in 12 rolls. We know:

• $n = 12$ (the number of trials)
• $p = \frac{1}{6}$ (the probability of success in each trial)

We want to find $P(X \geq 9)$, which can be expressed as:
$P(X \geq 9) = 1 - P(X \leq 8)$

Using the complementary cumulative distribution function for the binomial distribution, this is calculated as:

$P(X \geq 9) = 1 - \sum_{k=0}^{8} \binom{12}{k} \left(\frac{1}{6}\right)^{k} \left(\frac{5}{6}\right)^{12-k}$

Therefore, the correct expression for the required probability is captured in option A:
$P(\tilde{p} \geq \frac{3}{4})$