# Introductory Examples

On flipping a fair coin, one expects to get “heads” about half of the time, and “tails” about half of the time.   Try this with the attached random flip generator.   It will record up to five hundred attempts and show you the number of heads, the number of tails, and the fraction of the times each occurs.

Click here to open a copy of this so you can experiment with it. You will need to be signed in to a Google account.

On rolling a fair die, one expects to get each of the values   $1$   through   $6$   about one-sixth of the time.   Try this with the attached random flip generator.   It will record up to five hundred attempts and show you the number of times each of the six possible values occurs along with their fractional occurrence.

Click here to open a copy of this so you can experiment with it. You will need to be signed in to a Google account.

A shop opens every morning at 9am and observes how long it takes (in minutes) for the first customer to enter.   What follows is a spreadsheet record of this waiting time over five hundred openings.   The spreadsheet also computes bins (with   $w=\,$ waiting time, the bins are   $\, w\le 1{\rm m}\,$ , $\, 1{\rm m} \lt w \le 2{\rm m}\,$ , â€¦ , $\, 9{\rm m} \lt w \le 10{\rm m}\,$ , and $10{\rm m} \lt w\,$ ), and displays a histogram containing this information.

Click here to open a copy of this so you can experiment with it. You will need to be signed in to a Google account.

Each of these processes has separate occurrences of individual events.   Separate events seem to have nothing to do with one another.   And the results of the events seem to be random.   These intuitive notions are summed up in the the notion of probability, its rules and their consequences.   This chapter will present the essentials of the mathematical theory of probability.