Find minimum/maximum n for probability condition

4 In a certain country, on average one student in five has blue eyes.

1. For a random selection of \(n\) students, the probability that none of the students has blue eyes is less than 0.001 . Find the least possible value of \(n\).
2. For a random selection of 120 students, find the probability that fewer than 33 have blue eyes.

7 A factory makes water pistols, \(8 \%\) of which do not work properly.

1. A random sample of 19 water pistols is taken. Find the probability that at most 2 do not work properly.
2. In a random sample of \(n\) water pistols, the probability that at least one does not work properly is greater than 0.9 . Find the smallest possible value of \(n\).
3. A random sample of 1800 water pistols is taken. Use an approximation to find the probability that there are at least 152 that do not work properly.
4. Justify the use of your approximation in part (iii).

13 It is known that \(26 \%\) of adults in the UK use a certain app. A researcher selects a random sample of 5000 adults in the UK. The random variable \(X\) is defined as the number of adults in the sample who use the app. Given that \(\mathrm { P } ( X < n ) < 0.025\), calculate the largest possible value of \(n\).

7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.

1. In a local council, \(60 \%\) of households recycle at least half of their waste. A random sample of 80 households is taken.

The random variable \(X\) represents the number of households in the sample that recycle at least half of their waste.

1. Using a suitable approximation, find the smallest number of households, \(n\), such that $$\mathrm { P } ( X \geqslant n ) < 0.05$$ The number of bags recycled per family per week was known to follow a Poisson distribution with mean 1.5 Following a recycling campaign, the council believes the mean number of bags recycled per family per week has increased. To test this belief, 6 families are selected at random and the total number of bags they recycle the following week is recorded. The council wishes to test, at the 5\% level of significance, whether or not there is evidence that the mean number of bags recycled per family per week has increased.
2. Find the critical region for the total number of bags recycled by the 6 families.

1. A multiple choice examination paper has \(n\) questions where \(n > 30\)

Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
7.

1. In a large restaurant an average of 3 out of every 5 customers ask for water with their meal.

A random sample of 10 customers is selected.

1. Find the probability that
   1. exactly 6 ask for water with their meal,
   2. less than 9 ask for water with their meal. A second random sample of 50 customers is selected.
2. Find the smallest value of \(n\) such that $$\mathrm { P } ( X < n ) \geqslant 0.9$$ where the random variable \(X\) represents the number of these customers who ask for water.

5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).

1. \(\mathrm { E } ( X )\),
2. \(\operatorname { Var } ( X )\).
   1. Write down
   2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
   3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.