10.1 Key concepts

Probability in everyday life

Probability is connected with uncertainty. In any statistical experiment, the outcomes that occur may be known, but exactly which one might occur is usually not known. Mathematically, probability theory formulates incomplete knowledge pertaining to the likelihood of an occurrence. For example, a meteorologist might say there is a 60% chance that it will rain tomorrow. This means that in 6 of every 10 times when the world is in the current state, it will rain tomorrow.

In everyday speech, we often refer to probabilities using percentages between 0% and 100%. A probability of 100% (100 out of 100) means that an event is certain, whereas a probability of 0% (0 out of 100) is often taken to mean the event is impossible. However, in certain circumstances, there can be a distinction between logically impossible and occurring with zero probability if the sample space is large enough (i.e. infinite) or in some other way indeterminate. For example, in selecting a random fraction between 0 and 1, the probability of selecting 1/2 is 0, but it is not logically impossible (since there are infinitely many numbers), while trying to predict the exact number of raindrops to fall into a large enough area is also 0 (or close to it), since the possible number of raindrops that might occur at a given time is both very large and difficult to determine.

Another way of referring to probabilities is odds. The odds of an event is defined as the ratio of the probability that the event occurs to the probability that it does not occur. For example, the odds of a coin landing on a given side are $\frac{0.5}{0.5}=1$ , usually written "1 to 1" or "1:1". This means that on average, the coin will land on that side as many times as it will land on the other side.

The simplest example: equally likely outcomes

We say two outcomes are equally likely if they have an equal chance of happening. For example when a fair coin is tossed, each outcome in the sample space $S=\{heads,tails\}$ is equally likely to occur.

Probability is a function of events (since it is not possible to have a single event with two different probabilities occurring), so we usually denote the probability $P$ of some event $E$ occurring by $P\left(E\right)$ . When all the outcomes are equally likely (in any activity), it is fairly straightforward to count the probability of a certain event occuring. In this case,

$P\left(E\right)=n\left(E\right)/n\left(S\right)$

For example, when you throw a fair die the sample space is $S=\{1;2;3;4;5;6\}$ so the total number of possible outcomes $n\left(S\right)=6$ .

Event 1: The value of the topmost face of the die (call it 'k') is 4

The only possible outcome is a $4$ , i.e $E=\left\{4\right\}$ . So $n\left(E\right)=1$ .

Probability of getting a 4: $P(k=4)=n\left(E\right)/n\left(S\right)=1/6$ .

Event 2: The value of the topmost face is a number greater than 3

Favourable outcomes: $E=\{4;5;6\}$

Number of favourable outcomes: $n\left(E\right)=3$ .

Probability of getting a number greater than 3: $P(k>3)=n\left(E\right)/n\left(S\right)=3/6=1/2$ .