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Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?


At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

Concrete Calculation

Age 14 to 16
Challenge Level

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. The shape is designed to avoid subsidence that might be caused by the roots of some very large trees nearby. The hole is perfectly symmetrical with vertical walls, 1.2 metres high, surrounding a rectangular area measuring 6 metres by 5 metres. A mound of earth in the centre, the shape of the frustum of a pyramid, has a rectangular base 4.8 metres by 3.8 metres and a horizontal rectangular top 3.8 metres by 2.8 metres. The height of this mound is 0.9 metres. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations? Can you explain the method simply to help the builders to do the calculations for similar shaped rafts of other dimensions?