Mutually exclusive & independent events

A company sells furniture.

The company has a telephone number that customers can use if they want to complain. At the end of the call customers are asked if they are satisfied or not.

Originally the company had one call centre, Call centre A. The company now has another call centre, Call centre B.

Since Call centre B opened,

• when a call is answered the probability that it is answered by Call centre A is 0.6

• of those calls answered by Call centre A, 70% result in a customer being ‘satisfied’

• the probability of a call answered by Call centre B resulting in a customer being ‘satisfied’ is p

A call is picked at random.

Complete the probability tree diagram for this call.

Given that a customer is ‘satisfied’ the probability that the call was answered by Call centre B is 0.3

Find the probability of a call answered by Call centre B leading to a customer being satisfied.

The company sells tables that customers have to put together.

A table needs 4 connectors to hold it together. A bag of 4 connectors is supplied with each table.

The company previously bought the connectors from QuickC. The probability of a QuickC connector failing is 0.05 Connectors fail independently of each other.

A customer buys a table.

Work out the probability that the customer will need one or more replacement connectors because of the failure of QuickC connectors.

The company now buys the connectors from KlickT. The probability of a KlickT connector failing is 0.16

Dina thinks that putting two more connectors in the pack would reduce the probability that a customer will need one or more replacement connectors to less than it was when QuickC connectors were used.

Is Dina correct? You must show your working.

5 120 musicians were asked what instruments they play. The most popular instruments were piano (P), drums (D) and guitar (G).

10 musicians play all three instruments 13 musicians play both piano and guitar 17 musicians play both piano and drums 23 musicians play both guitar and drums 50 musicians play the piano 40 musicians play the drums 35 musicians play the guitar

Complete the Venn diagram using this information.

One musician is to be chosen at random.

Find P(G)

Find P(G|D)

Toga claims that playing the guitar and playing the drums are independent events.

Do you agree with Toga? Give a reason for your answer.

The Venn diagram shows information about 50 books on a reading list.

A is the event that the book is a science fiction book. B is the event that the book is an audio book.

The numbers in the Venn diagram show the number of books.

One of the books is chosen at random.

Find P(A).

Find P(A B).

Mike concludes that A and B are independent events.

Is Mike correct? Give a reason for your answer.

Two different events, C and D, are such that

Find P(C D).

There are 8 coloured discs in bag X and 9 coloured discs in bag Y.

In bag X, there are 3 red discs and 5 yellow discs. In bag Y, there are 5 red discs and 4 yellow discs.

Sonia takes at random one disc from bag X and she takes at random one disc from bag Y.

Calculate the probability that the two discs taken by Sonia are the same colour.

The mean of the daily winter temperatures and the mean of the daily summer temperatures in the UK have been recorded for each year from 1910 to 2018

This information has been used to model the two distributions of the mean temperatures.

The graphs below give information about these models.

Write down the name of the distribution that is suggested by each of these graphs.

Comment on the difference between the means of these two distributions.

The standard deviation for the distribution of the summer temperatures is 0.8 °C, correct to one decimal place.

Using the graph for the winter temperatures, calculate an estimate for the standard deviation of the distribution of the winter temperatures, correct to one decimal place.

The maximum height of the graph above the temperature axis for the winter temperatures is less than the maximum height of the graph above the temperature axis for the summer temperatures.

Explain why.

Carol says that the standard deviation of daily temperatures in summer for the years from 1910 to 2018 will also be 0.8 °C.

Explain whether or not Carol is correct.

Greta assumes that the means of the daily temperatures each year are independent.

She works out the probability that the means of the daily summer temperatures are greater than 15.6 °C for two consecutive years.

She concludes that this probability is less than 0.001

Using the model for the daily summer temperatures, assess Greta’s conclusion.