Conditional probability

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.

120 musicians were each asked if they play any of the instruments piano, guitar or drums.

40 musicians play the drums.

The Venn diagram gives information about the numbers of these musicians who play these instruments.

A musician is chosen at random.

Calculate the probability that the musician plays all three instruments.

Work out the value of x and the value of y

A musician is chosen at random.

Calculate P (plays the guitar | plays the piano)

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.

Olga is investigating the types of exercise taken by a group of 120 people. Olga asked each of these 120 people whether or not, for exercise, they go to a gym (G) or walk (W) or cycle (C).

The incomplete Venn diagram gives some information about Olga’s results.

One of the people that Olga asked is to be chosen at random.

Given that, for this person,

complete the Venn diagram.

A census found out information about people in Europe who were aged 15 to 19 in 2017

The Venn diagram gives information about the employment status and the education status of each of these people.

A is the event that the person is employed. B is the event that the person is in education.

The numbers in the Venn diagram are percentages of the total number of people in Europe who were aged 15 to 19 in 2017

(Source: https://ec.europa.eu/eurostat/)

What percentage of these people were not employed and were not in education?

One person aged 15 to 19 in 2017 is to be chosen at random.

Find P(B).

Find P(B A).

Using your answers to part (b) and to part (c), describe the effect that being employed has on the likelihood of being in education for these people. Justify your answer.

The manager of a garage records optional extra features sold with each new car sold last month. The optional extra features are air conditioning (A), bluetooth (B) and sat nav (S).

Last month, 80 cars were sold 7 cars were sold with sat nav and bluetooth and air conditioning 15 cars were sold with sat nav and bluetooth only 21 cars were sold with sat nav and air conditioning only 10 cars were sold with bluetooth and air conditioning only 8 cars were sold with sat nav only 33 cars in total were sold with bluetooth 44 cars in total were sold with air conditioning.

Complete the Venn diagram using this information.

Inge picks at random a car that was sold last month.

She thinks that the probability that the car has bluetooth given that it has sat nav is greater than the probability that the car has bluetooth given that it has not got sat nav.

Is Inge correct? You must show your working.

Second-hand cars are also sold at the garage. The manager records whether the second-hand cars have a full service history or not. She also records whether each second-hand car has a breakdown or not in the first year after being sold.

The table gives information about the second-hand cars sold at the garage in 2018

Paul says that the relative risk of a second-hand car without a full service history having a breakdown in the first year after being sold, compared with a second-hand car with a full service history having a breakdown in the first year after being sold, is 2

Show that Paul is correct.

Interpret a relative risk of 2 in this context.