Random procedures

We want to talk about the probabilities of events: things that may happen. This sounds impossibly broad and unwieldy, and it is. We therefore usually limit ourselves to a particular context in which possible outcomes are well defined and can be specified, at least in principle, beforehand. We also need to specify the procedure that is to be carried out, that will produce one of these outcomes.

We call such a phenomenon a random procedure or, alternatively, a random process. It need not be an experiment; all that is required is that the process for obtaining the outcome is well defined, and the possible outcomes are able to be specified. Why is the word `random' used? Its purpose is to indicate that the process is inherently uncertain: before we make the observation, we do not know which of the possible outcomes will occur, but we do know what is on the list of possible outcomes. Here are some examples:

  1. A standard 20 cent coin is tossed: a rapid spin is delivered by the thumb to the coin as it sits on the index finger; when it lands on the palm of the catching hand and is then flipped onto the back of the other hand, we observe the uppermost face. The possible outcomes are H (head) or T (tail).
  2. A fair six-sided die is shaken vigorously in a cup and rolled onto a table; the number of spots on the top side is recorded. The possible outcomes are \(1,2,3,4,5,6\).
  3. A lottery such as Powerball is observed and the number of the `Powerball' is recorded. The possible outcomes for Powerball are \(1,2,3,\dots,45\).

We start with these simple examples because probability statements associated with them have an obvious intuitive meaning. For example, it is commonly accepted that the chance of obtaining a head when you toss a coin equals \(\dfrac{1}{2}\), or 50%. Indeed, in many sports, a coin toss is used to determine who makes an initial choice that may involve an advantage, such as which end to kick to in football, whether to serve or receive in tennis, and whether to bat or bowl in cricket.

Similarly, when dice are used in board games, it is usually implicitly assumed that each possible outcome is equally likely, and therefore has probability \(\dfrac{1}{6}\).

In commercial lotteries such as Tattslotto and Powerball, it is a regulatory requirement that each outcome is equally likely.

But for the moment, we are just considering these examples as random procedures; we will come to the issue of probability later in this module. And we will see that there is more to it than these simple intuitions.

Here are some examples that are more complex:

  1. In a game of Tetris, a sequence of three consecutive pieces is observed.
  2. Five babies born in 1995 are followed up over their lives, and major health and milestone events are recorded. This really involves several random procedures; a specific one is considered later in this section.
  3. 273 students attempt Mathematical Methods in 2015. In June 2017, for each student, it is recorded whether or not he or she is a tertiary student.
  4. A new mobile phone is tested, and the time until its battery needs recharging is observed.

Example: Tetris

In Tetris there are seven distinct shapes made up of four squares. In serious Tetris discussions they are given letter labels which roughly correspond to their shapes. These are \(\mathtt{I}\) for the long stick, \(\mathtt{T}\) for the T-shape, and so on; the others are \(\mathtt{J}\), \(\mathtt{L}\), \(\mathtt{O}\), \(\mathtt{S}\) and \(\mathtt{Z}\).

What are the possible sequences of three consecutive pieces that we may observe?

We can list them alphabetically:

\[ \mathtt{III}, \ \mathtt{IIJ}, \ \mathtt{IIL}, \ \mathtt{IIO}, \ \mathtt{IIS}, \ \mathtt{IIT}, \ \mathtt{IIZ}, \ \mathtt{IJI}, \ \mathtt{IJJ}, \ \dotsc, \ \mathtt{ZZZ}. \]

There is a large number of possible sequences. How many?

Each position in the sequence can be any of the seven possible shapes, so the number of possible sequences is

\[ 7 \times 7 \times 7 = 7^3 = 343. \]

The previous example illustrates the multiplication rule, which has been covered informally in earlier years. This rule tells us that, if a sequence of \(k\) separate processes are considered, and the first can be done in \(n_1\) ways, the second in \(n_2\) ways, and so on, up to the \(k\)th process having \(n_k\) ways of being carried out, then the number of possible ways that the \(k\) processes can be carried out successively is \(n_1 \times n_2 \times \dots \times n_k\).

Example: Five people born in 1995

For five people born in 1995, we may record a number of different health, illness and injury phenomena, and usually a particular random procedure will focus on one of these. We might consider the specific injury status: `experienced a broken leg at least once prior to reaching age 20'. What are the possible outcomes for this random procedure?

Label the individuals with the letters \(A\) to \(E\) (the first five letters of the alphabet). We use the symbol \(A_0\) to indicate that individual \(A\) does not experience a broken leg prior to reaching age 20, and \(A_1\) to indicate that individual \(A\) does have this experience, and similarly for the other four individuals. One possible outcome is \(A_0 B_0 C_0 D_0 E_0\). In this outcome, none of the individuals has had a broken leg prior to turning 20. Each individual either does or does not have at least one break, so the total number of possible outcomes is \(2^{5} = 32\).

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