2022 RVHS J2 CT P1 Q11

To aid in contact tracing during the Covid-19 pandemic, the government of a country introduces the use of TraceWith App and Token for its people. Each individual in the population chooses to use either the App or the Token. On average, \(p\%\) of the population chose to use the TraceWith Token.

\(n\) individuals are randomly selected for a survey and the number of individuals who use the TraceWith Token is denoted by the random variable \(X\).

1. State in context, two assumptions needed for \(X\) to be well modelled by a binomial distribution.[2]
2. Given that \(X\) is modelled by a binomial distribution, \(\mathrm{E}\left( X \right)=3.125\) and \(\mathrm{Var}\left( X \right)=\mathrm{2}\mathrm{.734375}\), determine the values of \(n\) and \(p\).[3]

Assume that \(X\) has a binomial distribution \(\mathrm{B}\left( 35,0.2 \right)\).

3. Determine the most likely number of individuals who use the TraceWith Token.[2]
4. The probability that \(a\) or more individuals use the TraceWith Token is more than \(0.9\). Find the largest value of \(a\).[2]
5. Given that fewer than half of the surveyed individuals use the TraceWith Token, find the probability that more than \(10\) surveyed individuals use the token.[3]