Probability - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {21}Example: Election ballots (California)It is understood that there is a ‘positional bias’ in the voting behaviour of undecidedvoters in elections; that is, some of them tend to give their voting preference in the orderthat the candidates are listed on the ballot paper.For this reason, in California, about three months before an election, the Secretary ofState produces a random ordering of the letters of the alphabet. This is used to definethe order in which the names of candidates are printed on the ballot papers. What’smore, it is applied not just to the first letter of each candidate’s name, but to the otherletters in the names also.Example: PowerballThe commercial lottery Powerball has operated in Australia since 1996. There are 45balls that can appear as the Powerball. A probability model based on symmetry andrandom mixing implies that the chance of any particular ball appearing as the Powerballis 145≈ 0.022. (We are rounding to three decimal places here.) This is the idealised modelthat is intended on grounds of fairness for those who play the game, and we may believethat the model applies, to a very good approximation. There are laws governing lotteriesthat aim to ensure this.At the time of writing, there have been 853 Powerball draws. The following table showsthe relative frequencies of some of the 45 numbers from the 853 draws.Relative frequencies for the PowerballPowerball 1 2 3 4 5 ... 43 44 45Relativefrequency21 14 20 21 1519 22 22...853 853 853 853 853 853 853 8530.025 0.016 0.023 0.025 0.018 ... 0.022 0.026 0.026We may check assumptions about randomness by looking at the relative frequencies.We may ask: Are they acceptably close to the probability implied by a fair draw, whichis 0.022? Obviously, if we are to do this properly, we must look at the whole distribution,and not just the part shown here.