Small sample binomial probability

5 A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

1. Show that the probability that the spinner lands on the blue side is \(\frac { 1 } { 8 }\).
2. The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.
3. The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.

2. Crispy-crisps produces packets of crisps. During a promotion, a prize is placed in \(25 \%\) of the packets. No more than 1 prize is placed in any packet. A box contains 6 packets of crisps.

1. 1. Write down a suitable distribution to model the number of prizes found in a box.
   2. Write down one assumption required for the model.
2. Find the probability that in 2 randomly selected boxes, only 1 box contains exactly 1 prize.
3. Find the probability that a randomly selected box contains at least 2 prizes. Neha buys 80 boxes of crisps.
4. Using a normal approximation, find the probability that no more than 30 of the boxes contain at least 2 prizes.

3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently. A random sample of 10 of these plates is selected.

1. Find the probability that the sample contains
   1. fewer than 2 flawed plates,
   2. at least 6 flawed plates.
      (4) George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
2. Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that $$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$ where \(F \sim \mathrm {~B} ( 120,0.45 )\) The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
3. Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}

1. A manufacturer produces large quantities of coloured mugs. It is known from previous records that \(6 \%\) of the production will be green.

A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample.
2. Find the probability that there were exactly 3 green mugs in the sample. A random sample of 125 mugs was taken.
3. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation,
   2. a Normal approximation.

3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.

1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
2. Specify a suitable sampling frame and explain how such a sample could be taken.
   (2 marks)
   The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
   (6 marks)
4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).

5

1. In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random. Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
2. 1. In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people. A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
   2. In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people. A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
   3. The random variable \(T\) denotes the total number of people in the two selected towns who have this blood disorder. Write down the distribution of \(T\) and hence determine \(\mathrm { P } ( T > 16 )\).