2.04c Calculate binomial probabilities

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.

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 \(k\).

14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.

1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.

The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.

1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm. [2]
2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]

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]

A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to

1. 1. show that P\((J > 510) = 0.3446\) [2]
   2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]

Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.

1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]

Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)

1. find the value of \(r\) and the value of \(k\) [7]

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]

In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].

1. Write down P(T < 0.2). [1]
2. Write down E(T). [1]
3. Use integration to find Var(T). [4]

A group of 20 children each play this game once.

1. Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]

The children are allowed to practise this game so that this continuous uniform model is no longer applicable.

1. Explain how you would expect the mean and variance of T to change. [2]

It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by $$g(t) = \begin{cases} 4t, & 0 \leq t \leq 0.5, \\ 4 - 4t, & 0.5 \leq t \leq 1, \\ 0, & otherwise. \end{cases}$$

1. Using this model show that P(T < 0.2) = 0.08. [2]

A group of 75 experienced players each played this game once.

1. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]

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 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]

The random variable R has the binomial distribution B(12, 0.35).

1. Find P(R ≥ 4). [2]

The random variable S has the Poisson distribution with mean 2.71.

1. Find P(S ≤ 1). [3]

The random variable T has the normal distribution N(2.5, 5²).

1. Find P(T ≤ 18). [2]

The random variable \(R\) has the binomial distribution B(12, 0.35).

1. Find P(\(R \geq 4\)). [2]

The random variable \(S\) has the Poisson distribution with mean 2.71.

1. Find P(\(S \leq 1\)). [3]

The random variable \(T\) has the normal distribution N(25, \(5^2\)).

1. Find P(\(T \leq 18\)). [2]

1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]

A researcher has suggested that 1 in 150 people is likely to catch a particular virus. Assuming that a person catching the virus is independent of any other person catching it,

1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]

A factory produces components of which 1\% are defective. The components are packed in boxes of 10. A box is selected at random.

1. Find the probability that the box contains exactly one defective component. [2]
2. Find the probability that there are at least 2 defective components in the box. [3]
3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]

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]

A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5\% level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly. [6]

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]