Binomial distribution

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.

Peter drops his children off at a breakfast club every Monday morning and every Tuesday morning. He either drops them off on time or late.

The probability that Peter drops his children off on time on a Monday morning is three times the probability that he drops them off late on a Monday morning.

The probability that Peter drops them off on time on a Monday morning and on a Tuesday morning is 0.6

The probability that Peter drops them off late on a Tuesday morning given that he has dropped them off late on a Monday morning is 0.3

Complete the tree diagram below.

Peter drops his children off at the breakfast club every Monday morning and Tuesday morning over a 4-week period.

Assume that the probability of dropping them off on time on both a Monday morning and a Tuesday morning remains constant and that each week is independent.

Work out the probability that, over a 4-week period, he drops his children off on time on a Monday morning and on time on a Tuesday morning on 3 or more occasions.

A game is played with 4 ordinary 6-faced dice. Each dice is to be rolled once and the number of dice that land on a six is recorded.

Write down two conditions needed so that a binomial distribution is a suitable model for the number of sixes recorded.

Calculate the probability that all of the 4 dice land on a six. Give your answer as a fraction.

Calculate the probability that at least 2 of the 4 dice land on a six. Give your answer as a fraction.

A company contracts two manufacturers, A and B, to make an item that the company sells.

Manufacturer A makes 65% of the items for the company to sell.

It is found that 9% of the items made by manufacturer A do not meet the company’s required standard.

It is found that 12% of the items made by manufacturer B do not meet the company’s required standard.

An item is to be picked at random.

Complete the probability tree diagram for this item.

One of the items that the company is to sell is picked at random.

M is the event that the item was made by manufacturer A. N is the event that the item does not meet the required standard.

Find P(MN).

The company is planning to sell multipacks containing 6 items made by manufacturer A.

The financial director of the company plans to use the binomial distribution to model the number of items in the multipack not meeting the required standard.

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

A multipack made by manufacturer A is to be selected at random.

Work out the probability that this multipack will have 2 or more items that do not meet the required standard. Give your answer correct to 3 decimal places.

Fruitees sweets come in different flavours.

There are 8 sweets in a pack of mixed flavours and the flavours for each pack are chosen at random.

The mean number of strawberry flavour Fruitees in a pack of 8 sweets is 2

Ed suggests that the number of strawberry flavour Fruitees in a pack of 8 sweets can be modelled by a binomial distribution.

By considering the conditions that make a binomial distribution a suitable model, explain why Ed’s suggestion is appropriate.

One sweet is selected at random from a pack of Fruitees.

Find the probability that the flavour of this sweet is strawberry.

Ed buys a pack of Fruitees.

Find the probability that there will be exactly 3 strawberry flavour Fruitees in the pack.