Probability - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {43}cdefghWith a sufficiently vigorous and lengthy shake, the mixing is random, leading to nopreference for any outcome, assuming a symmetric die.An overhand shuffle needs to be done for a very long time to produce randomnessin the order of the pack, relative to the starting order. If the shuffle is done just a fewtimes, traces of the original order will persist.A riffle shuffle produces a random order much more effectively than an overhandshuffle.It seems clear that this blast of air produces such chaotic and extensive mixing of theballs that, assuming the balls are of uniform size, shape and weight, all outcomes willbe (close to) equally likely.Historically, random number generators have been of varying quality; it depends onthe method used. Most computer games now use algorithms that will not producedetectable non-randomness.A simple physical device like this may well have inadequate mixing.Exercise 6What you get depends on the specific coin. Were you surprised by your results?Exercise 7There are indeed five possibilities for the number of aces in the hand. But these differentpossibilities are not equally likely. A standard deck has 52 cards. There are thousandsof equiprobable distinct hands of four cards that can be dealt: 270 725, to be precise.Only one of these possible hands contains all 4 aces, whereas there are many hands thatcontain no aces. So the chance of four aces isno aces is much greater.1270 725, while the probability of a hand withExercise 8There is some simple reasoning to be applied here. Consider the ballot for the first halfof the year, in which 29 February could be selected. In most years, no men born on29 February could be recruited, even if that marble came up, since their birth year wasnot a leap year (for example, in 1965 the birth year was 1945). In years such as 1968,the men turning 20 were born in the leap year of 1948, and some would have been bornon 29 February. For those men born on 29 February 1948, their chances of conscriptionwere the same as other men born in the first half of 1948.In fact, there were only two relevant leap years, 1948 and 1952, involving conscription inthe years 1968 and 1972. The date 29 February came up in the selected marbles in 1972and not in 1968.