Other uses of simulation

Simulations can help us to answer questions about a variety of other models (or populations). The following example was suggested by Kay Lipson from Swinburne University of Technology in Australia.

Does Australia Post deliver on time?

The Herald-Sun newspaper published the following article on November 25 1992.

Doubt has been cast over Australia Post's claim of delivering 96 per cent of standard letters on time.

A survey conducted by the Herald-Sun in Melbourne revealed that less than 90 per cent of letters were delivered according to the schedule.

Herald-Sun staff posted 59 letters before the advertised...

Campbell Fuller, Herald-Sun, 25 November 1992.

Is the author justified in disputing Australia Post's claim that 96% of letters are delivered on time?

A simulation

If Australia Post's claim is correct, and every letter independently has probability 0.96 of being delivered on time, we know that the number delivered on time out of 59 letters will be a random quantity. From the information in the article, we can deduce that 52 out of the Herald-Sun's 59 letters arrived on time (a proportion 52/59 = 0.881).

How unlikely is it to get as few as 52 out of 59 letters arriving on time if Australia Post's claim that the probability of letters arriving on time is 0.96 is correct?

A simulation helps to answer this question.

Click Simulate to randomly 'deliver' 59 letters, with each independently having probability 0.96 of arriving on time. Click Accumulate then run the simulation between 100 and 200 times. (Hold down the Simulate button to speed up the process.)

Observe the distribution of the number of letters arriving on time. The proportion of simulations with 52 or fewer letters arriving on time is shown to the right of the dot plot. Observe that this rarely happens.

We therefore conclude that the article is justified — only 52 letters being delivered on time is most unlikely if Australia Post's claim is correct.

We will return to this example later.