2023 RI P2 Q10

Two fair dice are thrown and the product of the two numbers on the top faces is computed. Let \({{R}_{n}}\) be the remainder when the product is divided by \(n\).

1. Find the probability distribution of \({{R}_{2}}\) and hence find \(\mathrm{E}\left( {{R}_{2}} \right)\).[3]
2. Show that \(\mathrm{P}\left( {{R}_{6}}=0 \right)=\frac{5}{12}\).[1]

The two dice are thrown m times and the product of the two numbers on the top faces are computed each time. \(X\) is the number of products that are divisible by \(6\) after \(m\) throws.

3. Show that \(\mathrm{P}\left( X=2 \right)=Am\left( m-1 \right){{\left( \frac{7}{12} \right)}^{m}}\), where \(A\) is a constant to be determined. Given that x\(\mathrm{P}\left( X=2 \right)=\mathrm{0}\mathrm{.03604}\), correct to \(5\) decimal places, deduce the value of \(m\), justifying your answer.[4]
4. Class \(C\) has \(35\) students and class \(D\) has \(30\) students. Each student in the two classes throws the two dice \(10\) times. Estimate the probability that the average number of products that are divisible by \(6\) obtained by a student in class \(C\) exceeds the average number of products that are divisible by \(6\) obtained by a student in class \(D\) by more than \(0.2\).[5]