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# Binomial or Not?

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### Over-booking

### Statistics - Maths of Real Life

### Binomial Conditions

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Age 16 to 18

Challenge Level

Which of these situations could be modelled using the binomial distribution?

For those which can be modelled by the binomial distribution $\mathrm{B}(n,p)$, what are "success" and "failure", and what are $n$ and $p$?

For those which are not, why not? Is there something small you could change that would make the binomial distribution appropriate? Is the binomial distribution at least a good approximation in this situation?

*This resource is part of the collection Statistics - Maths of Real Life*

For those which can be modelled by the binomial distribution $\mathrm{B}(n,p)$, what are "success" and "failure", and what are $n$ and $p$?

For those which are not, why not? Is there something small you could change that would make the binomial distribution appropriate? Is the binomial distribution at least a good approximation in this situation?

- A bag contains red, green and blue balls. 10 balls are taken at random, one at a time. Each ball's colour is recorded, and then returned to the bag.

- A coin is flipped until a tail is obtained. The total number of flips needed is recorded.

- The number of rainy days in April in the village of Springfield is recorded.

- The children in a class each do the same mathematics test. The number who score above 80% is recorded.

- Five fair coins are stuck to a piece of clear plastic. The plastic is flipped in the air, and the number of heads showing when the plastic lands is recorded.

- A person plays a lottery every week. They record the number of times they win a prize during one year.

- A cancer drug is being tested. 1000 patients are given the drug, and the number of patients who die within five years is recorded.

- A basketball player is practising taking shots. The number of successful shots out of 10 attempts is recorded.

- A bag contains red and blue balls. 10 balls are taken at random, one at a time. Each ball's colour is recorded, and then returned to the bag.

- A box of pens contains working pens and broken pens. 10 pens are taken together from the box at random, and the number of working pens is recorded.

- A bag contains red, green and blue balls. 10 balls are taken from the bag at random one at a time, and replaced immediately. The number of green balls taken is recorded.

- A farmer is planting a crop. On average, a certain percentage of the seeds grow to maturity. The number of seeds that grow to maturity in this field in this year is recorded.

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

This pilot collection of resources is designed to introduce key statistical ideas and help students to deepen their understanding.

When is an experiment described by the binomial distribution? Why do we need both the condition about independence and the one about constant probability?