2020 MI P2 Q9

Always Cool Company produces reusable ice pack gels. It is known that, on average, \(p\%\) of the reusable ice pack gels leak. Each day the quality control department takes a random sample of \(120\) reusable ice pack gels and checks them for leakage. The number of reusable ice pack gels that leak in a sample is denoted by the random variable \(X\).

1. State, in the context of the question, two assumptions needed for \(X\) to be well modelled by a binomial distribution.[2]

Assume now that \(X\) can be modelled by a binomial distribution,

2. It is given that the most likely number of reusable ice pack gels that leak in a sample is \(3\). Find the range of values of \(p\).[3]

For the rest of the question, the value of \(p\) is taken to be \(3\).

3. Find the probability that the quality control department finds at most \(4\) reusable ice pack gels that leaks in a sample.[1]
4. If the quality control department found at least \(5\) reusable ice pack gels that leak in a sample, find the probability that there are fewer than \(9\) ice pack gels that leak in a sample.[3]
5. The number of reusable ice pack gels that leak in a sample is independent of that of other samples. Find the probability that, in \(10\) randomly chosen samples, the eighth sample is the third sample with at most \(4\) reusable ice pack gels that leak.[3]