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Statistics, EXT1 S1 2023 HSC 2 MC

A standard six-sided die is rolled 12 times.

Let \(\hat{p}\) be the proportion of the rolls with an outcome of 2.

Which of the following expressions is the probability that at least 9 of the rolls have an outcome of 2?

  1. \(P\left(\hat{p} \geq \dfrac{3}{4}\right)\)
  2. \(P\left(\hat{p} \geq \dfrac{1}{6}\right)\)
  3. \(P\left(\hat{p} \leq \dfrac{3}{4}\right)\)
  4. \(P\left(\hat{p} \leq \dfrac{1}{6}\right)\)
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Let}\ X =\ \text{number of rolls}\ \geq 2\)

\(X \sim\ \text{Bin}\Big(12, \dfrac{5}{6}\Big)\)

\(\text{If}\ \ X=9\ \ \Rightarrow\ \ \hat{p} = \dfrac{9}{12}=\dfrac{3}{4} \)

\(\therefore P(X \geq 9) = P\Big(\hat{p} \geq \dfrac{3}{4}\Big) \)

\(\Rightarrow A\)