Probability Statistics 1

When is a die fair? You’ve rolled a die 100 times. The number 6 has appeared 23 times, more frequently then any of the other numbers. At one stage, it turns up three times in a row, resulting in you winning a board game. Your opponent claims that it isn’t a fair die. 2Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Questions to consider a)If a fair die is rolled 100 times, how many 6s do you expect? b)Would it be unusual for a 6 to be rolled 23 times in 100 rolls? c)Would it be surprising to roll three 6s in a row at some point? 3

Law of large numbers In 1689, the Swiss mathematician Jacob Bernoulli proved that as the number of trials increases, the proportion of occurrences of any given outcome approaches a particular number (such as 1/6). 4Source: [14 July 2011]

Fair die You’ve rolled three 6s in a row on a fair die. Are you a)more likely to roll a 6 b)less likely to roll a 6 c)just as likely to roll a 6 on your next go than a 5? 5

Independent events With many random phenomena, such as outcomes of rolling a die, what happens on previous trials has no effect on the next trial The trials are independent of each other 6

Gambling If you have lost many bets in a row, don’t assume that you are due to win if you continue to gamble The law of large numbers only guarantees long-run performance 7