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

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A standard six-sided die is rolled 12 times. Let $ ilde{p}$ be the proportion of the rolls with an outcome of 2. Which of the following expressions is the probabil... show full transcript

Worked Solution & Example Answer:A standard six-sided die is rolled 12 times - HSC - SSCE Mathematics Extension 1 - Question 2 - 2023 - Paper 1

Step 1

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

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Answer

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=16p = \frac{1}{6}.

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

  • n=12n = 12 (the number of trials)
  • p=16p = \frac{1}{6} (the probability of success in each trial)

We want to find P(X≥9)P(X \geq 9), which can be expressed as:
P(X≥9)=1−P(X≤8)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≥9)=1−∑k=08(12k)(16)k(56)12−kP(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(p~≥34)P(\tilde{p} \geq \frac{3}{4})

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