Verify conditions in context

5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).

1. State two conditions that are required for \(X\) to have a binomial distribution.
   On average, Hebe completes 7 out of 10 of these puzzles.
2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.

7 On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .

1. State two conditions needed for X to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
2. Find \(\mathrm { P } ( \mathrm { X } \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.

1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.

1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
2. 1. the probability that all 50 people turn up for the journey,
   2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
3. Find the bus company's expected total earnings per journey.

2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.

1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
2. Find \(\mathrm { P } ( T = 5 )\)
3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.

1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.

1. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.

Peta models \(H\) as \(\mathrm { B } ( 10,0.1 )\)

1. State two assumptions Peta needs to make to use her model.
2. Using Peta's model, find \(\mathrm { P } ( H \geqslant 4 )\) For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
3. find \(\mathrm { P } ( F = 5 )\) Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm { P } ( F = n )\) as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where \(\alpha\) is a constant.
4. Find the value of \(\alpha\)
5. Using Thomas' model, find \(\mathrm { P } ( F = 5 )\)
6. Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.

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.

1. State two conditions under which a random variable can be modelled by a binomial distribution. [2]

In the production of a certain electronic component it is found that 10% are defective. The component is produced in batches of 20.

1. Write down a suitable model for the distribution of defective components in a batch. [1]

Find the probability that a batch contains

1. no defective components, [2]
2. more than 6 defective components. [2]
3. Find the mean and the variance of the defective components in a batch. [2]

A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.

1. Using a suitable approximation, find the probability that the supplier will receive a refund. [4]

1. Briefly describe the main features of a binomial distribution. [2 marks]

I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52.

1. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\text{B}(10, \frac{1}{4})\). [2 marks]

After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\text{B}(10, \frac{1}{4})\), find

1. the probability of getting no hearts, [3 marks]
2. the probability of getting 4 or more hearts. [2 marks]
3. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn. [2 marks]

At a fairground, Kirsty throws \(n\) balls in order to try to knock coconuts off their stands. Any coconuts she knocks off are replaced before she throws again. Kirsty counts the number of coconuts she successfully knocks off their stands. On average, she knocks off a coconut with 20\% of her throws.

1. What assumptions are needed in order to model this situation with a binomial distribution? Explain whether these assumptions are reasonable. [2]

Kirsty uses a spreadsheet to produce the following diagrams, showing the probability distributions of the number of coconuts knocked off their stands for different values of \(n\). \includegraphics{figure_3}

1. Describe two ways in which the distribution changes as \(n\) increases. [2]

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 B(20, 0.13). [2]
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [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 B(20, 0.13). [2]
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [4]