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?

Escalator

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?

Spot the Card

Age 14 to 16 Challenge Level:

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The Trick:

Select any 15 cards from a pack and lay them out on the table - face up - in three equal columns.

Ask someone to choose one card from the 15 and tell you which column it is in.
• Collect up the three columns into piles keeping the cards in order.
• Put the three piles together with the chosen column in the middle.

Now lay out the cards again - face up - in three columns placing one card in each column in turn. Ask the person which column contains the card this time.

Collect up the cards in the same way and lay them out for a third time.

Ask once more which column the card is in - collecting the cards for the third and last time in the same way - with the chosen column in the middle.

Now count down to the middle (8th) card and show it to the person. "Is that your card?"

The problem:

Why does this trick always work?

Are there any other numbers of cards for which this works?

What is the largest number?

Can the trick be adapted for different numbers?