2.04b Binomial distribution: as model B(n,p)

5 A garden centre grows a particular variety of plant for sale. They sow 3 seeds in each pot and there are 6 pots in a tray. The probability that a seed germinates is 0.7 , independently of any other seeds.

1. State the probability distribution of the number of seeds in a pot that germinate.
2. Find the probability that, in a randomly chosen pot,
   1. exactly 2 seeds germinate,
   2. at least 1 seed germinates. After the seeds have germinated and become seedlings, some are removed (and discarded) so that there remains at most 1 seedling per pot.
   3. Write out the probability distribution of the number of seedlings per pot that remain.
   4. Find the probability that there is a seedling in every one of the 6 pots in a randomly chosen tray.

4 A survey into left-handedness found that 13\% of the population of the world are left-handed.

1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution \(\mathrm { B } ( 20,0.13 )\).
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than \(13 \%\) of the 20 children are left-handed.

5 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.

1. Construct a probability distribution table for \(X\).
2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.

4 On a particular day at a busy international airport, 75\% of the scheduled flights depart on time. A random sample of 16 flights is chosen.

1. Find the expected number of flights that depart on time.
2. For these 16 flights, find the probability that fewer than 14 flights depart on time.
3. For these 16 flights, the probability that at least \(k\) flights depart on time is greater than 0.9 . Find the largest possible value of \(k\).

Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.

1. Show that the probability of obtaining exactly one head is 0.225. [3]
2. Complete the following probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4
   P(\(X = x\))	0.05	0.225			0.075

3. Given that E(\(X\)) = 2.1, find the value of Var(\(X\)). [2]

80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

1. Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
2. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
3. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]

A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. [3]

In Marumbo, three quarters of the adults own a cell phone.

1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
3. Justify the use of your approximation in part (ii). [1]

At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.1. The hospital's business model assumed that this probability will be reduced. They wish to test whether this probability is now less than 0.1. A random sample of 50 appointments is selected and the number of patients that did not arrive is noted. This figure is used as a test statistic at the 5\% significance level.

1. Explain why this test is a one-tailed test and state suitable null and alternative hypotheses. [2]
2. Use a binomial distribution to find the critical region and find the probability of a Type I error. [5]
3. In fact 3 patients out of the 50 did not arrive. State the conclusion of the test, explaining your answer. [2]

The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.

1. Identify one potential problem with this sampling frame. [1]

Customers are asked to complete a survey about the quality of service they receive. Past information shows that 35\% of customers complete the survey. A random sample of 20 customers is taken.

1. Write down a suitable distribution to model the number of customers in this sample that complete the survey. [2]
2. Find the probability that more than half of the customers in the sample complete the survey. [2]

Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.

1. 1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
   2. Evaluate your expression, giving your answer to 3 significant figures. [3]
2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]

A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 20\% of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3, [2]
2. fewer than 2. [2]

One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,

1. use a suitable approximation to find the probability that there are enough first class stamps, [7]
2. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid. [1]

An airline knows that overall 3\% of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.

1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. [2]

By using a suitable approximation, find the probability that

1. more than 196 passengers turn up for this flight, [3]
2. there is at least one empty seat on this flight. [2]

The owner of a small restaurant decides to change the menu. A trade magazine claims that 40\% of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners X who choose organic foods. [2]
2. Find P(5 < X < 15). [4]
3. Find the mean and standard deviation of X. [3]

The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly. [5]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05. Eggs are packed in boxes of 12. Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that 40\% of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine. [1]
2. Suggest a suitable sampling frame for the survey. [1]
3. Identify the sampling units. [1]
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. [2]

As a pilot study the editor took a random sample of 25 subscribers.

1. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. [3]

In fact only 6 subscribers agreed to the name being changed.

1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the percentage agreeing to the change is less that the editor believes. [5]

The full survey is to be carried out using 200 randomly chosen subscribers.

1. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. [7]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable X, the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of X. [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(X) and Var (X). [3]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. the first 5 will occur on the sixth throw, [8]
2. in the first eight throws there will be exactly three 5s.

A disease occurs in 3\% of a population.

1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution. [2]
2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
3. Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

1. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]

The continuous random variable \(X\) is uniformly distributed over the interval \([-1,3]\). Find

1. E(\(X\)) [1]
2. Var(\(X\)) [2]
3. E(\(X^2\)) [2]
4. P(\(X < 1.4\)) [1]

A total of 40 observations of \(X\) are made.

1. Find the probability that at least 10 of these observations are negative. [5]