4.1 Probability topics: terminology

Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin twice is an example of an experiment.

The result of an experiment is called an outcome . A sample space is a set of all possible outcomes. Three ways to represent a sample space are to list the possible outcomes, tocreate a tree diagram, or to create a Venn diagram. The uppercase letter $S$ is used to denote the sample space. For example, if you flip one fair coin, $\mathrm{S\; =\; \{H,\; T\}}$ where $H$ = heads and $T$ = tails are the outcomes.

An event is any combination of outcomes. Upper case letters like $A$ and $B$ represent events. For example, if the experiment is to flip one fair coin, event $A$ might be getting at most one head. The probability of an event $A$ is written $\mathrm{P(A)}$ .

The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between 0 and 1, inclusive (includes 0 and 1 and all numbers between these values). $\mathrm{P(A)\; =\; 0}$ means the event $A$ can never happen. $\mathrm{P(A)\; =\; 1}$ means the event $A$ always happens. $\mathrm{P(A)\; =\; 0.5}$ means the event $A$ is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative fequency of heads approaches 0.5 (the probability of heads).

Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair , six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head(H) and a Tail(T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event $A$ when all outcomes in the sample space are equally likely , count the number of outcomes for event A and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, thesample space is $\mathrm{\{HH,\; TH,\; HT,\; TT\}}$ where $T$ = tails and $H$ = heads. The sample space has four outcomes. $A$ = getting one head. There are two outcomes $\mathrm{\{HT,\; TH\}}$ . $\mathrm{P(A)\; =}\frac{2}{4}$ .

Suppose you roll one fair six-sided die, with the numbers {1,2,3,4,5,6} on its faces. Let event $E$ = rolling a number that is at least 5. There are two outcomes $\mathrm{\{5,\; 6\}}$ . $\mathrm{P(E)\; =}\frac{2}{6}$ . If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, 2/6 of the rolls would result in an outcome of "at least 5". You would not expect exactly 2/6. The long-term relative frequency of obtaining this result would approach the theoretical probability of 2/6 as the number of repetitions grows larger and larger.