Probability - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {19}defghAn ‘overhand’ shuffle of cards; this is the shuffle that we usually first learn.A ‘riffle’ shuffle of cards: in this shuffle, two piles of cards are rapidly and randomlyinterwoven by gradual release from the two thumbs.The blast of air blown into the sphere containing the balls in Powerball.The random number generator that produces the next Tetris shape.The hat used for drawing raffles at the footy club.LinkFor a guide to various card shuffles (including the overhand and riffle shuffles), seewww.pokerology.com/poker-articles/how-to-shuffle-cards/There are also more subtle aspects of symmetry to be considered. In the case of a conventionaldie, the shape may be cubical and hence symmetrical. But is that enough? We alsoneed uniform density of the material of the cube, or at least an appropriately symmetricdistribution of mass (some dice are hollow, for example). If a wooden die is constructedwith a layer of lead hidden under one face, this would violate the assumption that wemake in assigning equal probabilities to all six outcomes.What about coin tossing and symmetry? We may assume that the coin itself is essentiallya cylinder of negligible thickness made from uniform density metal with two perfectlyflat sides. No actual coin is like this. The faces of coins have sculpted shapes on themthat produce the images we see. These shapes are clearly not actually symmetric. Howmuch does this matter?Exercise 6Use an ordinary 20 cent coin to carry out the followingexperiment. Find the flattest surface you can. Startingwith the coin on its edge, spin the coin fast; usually thisis most effectively done by delivering the spin from theindex finger of one hand on one side of the coin againstthe thumb of the other hand on the other side, as shownin the photo.The spin needs to be uninterrupted by other objects andstay on the flat surface throughout. There is a bit of skillrequired to get the coin to spin rapidly; practise until youcan get it to spin for about ten seconds before comingto rest.Starting position fornon-standard spin.Once you are adept at this technique, on a given spin you can save time near the end ofthe spin: when it is clear what the result will be, you can stop the motion and record it.Do this 30 times. How many heads and tails do you get?