Probability Distributions (Edexcel GCSE Statistics: Higher)

Isobel has collected data about the times taken by a group of students to solve a puzzle.

The table gives some of the percentiles of her data.

Isobel plans to use the mean and standard deviation of the times of these students to summarise her data.

Give a reason why Isobel's plans are appropriate.

Isobel claims that the times of these students can be modelled using a normal distribution with mean 6.7 minutes and standard deviation 0.82 minutes.

Assess whether or not the data support Isobel's claim.

A particular type of chocolate bar comes in a packet of 10. Each individual chocolate bar inside the packet has a coloured wrapper. The colour of the wrapper for each chocolate bar is assigned at random.

The mean number of chocolate bars with a purple wrapper in a pack of 9 bars is 4.

Vivienne suggests that the number of chocolate bars with purple wrappers in a pack of 9 chocolate bars can be modelled by a binomial distribution.

Consider the conditions required for a binomial distribution and explain why Vivienne's suggestion is appropriate.

One chocolate bar is selected at random from a packet of 9 chocolate bars.

Find the probability that the colour of its wrapper is purple.

Simon buys a pack of 9 chocolate bars.

Find the probability that exactly 5 of the chocolate bars will have a purple wrapper.

A manufacturer makes chairs. The probability that a chair does not pass inspection is 5 %, the probability that a chair does pass inspection is 95 %.

The manufacturer is planning to sell chairs in sets of 6 and wants to model the the number of chairs passing inspection using a binomial distribution.

Write down one condition that needs to be assumed so that the binomial distribution is an appropriate model to use.

A set of 6 chairs is to be selected at random.

Work out the probability that the set will contain at least 5 chairs that pass inspection.
Give your answer correct to 3 decimal places.

An activity consists of rolling 5 fair 6-sided dice and the number of dice landing on an odd value is recorded.

Calculate the probability that all of the 5 dice land on an odd value.
Give your answer as a fraction.

Calculate that at least 2 of the 5 dice land on an odd number.
Give your answer as a fraction.

A large sample of films are classified as either blockbusters or as independent films.

Production costs for independent films have a mean of $ 23 million and a standard deviation of $ 4 million.

Production costs for independent films can be modelled by a normal distribution.

Nadine says,

"Less than 85 % of independent films have production costs of between $ 11 million and $ 27 million"

Use statistical calculations to assess Nadine's conclusion.

The diagram below shows a sketch of the distribution of production costs of the blockbusters in the sample.

Explain why it is not appropriate to model the production costs of blockbusters using a normal distribution.

On the same grid, sketch a diagram showing the distribution of production costs for the independent films.

Vinnie believes that he has a biased dice, where the probability of rolling a six is 0.3.

In an experiment, Vinnie rolls the dice 4 times in a row.

He then repeats this experiment 200 times.

The table shows information about the number of sixes rolled for each experiment.

Determine whether or not the model is suitable for Vinnie's experiment.

The mean of the daily rainfall during the spring and the mean of the daily rainfall during the winter in a country have been recorded each year for 30 years.

The graphs below show the distributions of the mean rainfall in the spring and winter months.

The standard deviation for the distribution of the winter rainfall is 3 mm.

Oscar says that the standard deviation of the daily rainfall for the past 30 years will also be 3 mm.

Explain whether or not Oscar is correct.

Eunice assumes that the means of the daily rainfall each year are independent.

She works out the probability that the means of the daily winter rainfall are greater than 47 mm for 3 consecutive years.

She concludes that this probability is greater than 0.92.

Using the model for the daily winter rainfall, assess Eunice's conclusion.

The table below gives information about the distribution of eye colour in a particular population

It shows the proportion of people with blue, brown, green or hazel eyes.

8 people from the population are randomly selected.

(i) Name the probability distribution that can be used to model the number of people from these 8 people who will have blue eyes.

[1]

(ii) Write down one condition needed so that this distribution is a suitable model.

[1]

Work out the probability that exactly 1 person from these 8 people will have blue eyes.

On a different occasion, people from the population are randomly selected.

The probability that at least one of these people will have blue eyes is greater than 80 %.

What can you conclude, if anything, about the value of ?
You must show your working.