Probability Key Definitions Help

Probability Key Definitions—Event Versus Outcome

Here are definitions of some common terms that will help us understand what we are talking about when we refer to probability.

Event Versus Outcome

The terms event and outcome are easily confused. An event is a single occurrence or trial in the course of an experiment. An outcome is the result of an event.

If you toss a coin 100 times, there are 100 separate events. Each event is a single toss of the coin. If you throw a pair of dice simultaneously 50 times, each act of throwing the pair is an event, so there are 50 events.

Suppose, in the process of tossing coins, you assign ''heads'' a value of 1 and ''tails'' a value of 0. Then when you toss a coin and it comes up ''heads,'' you can say that the outcome of that event is 1. If you throw a pair of dice and get a sum total of 7, then the outcome of that event is 7.

The outcome of an event depends on the nature of the hardware and processes involved in the experiment. The use of a pair of ''weighted'' dice produces different outcomes, for an identical set of events, than a pair of ''unweighted'' dice. The outcome of an event also depends on how the event is defined. There is a difference between saying that the sum is 7 in a toss of two dice, as compared with saying that one of the dice comes up 2 while the other one comes up 5.

Sample Space

A sample space is the set of all possible outcomes in the course of an experiment. Even if the number of events is small, a sample space can be large.

If you toss a coin four times, there are 16 possible outcomes. These are listed in Table 3-1, where ''heads'' = 1 and ''tails'' = 0. (If the coin happens to land on its edge, you disregard that result and toss it again.)

If a pair of dice, one red and one blue, is tossed once, there are 36 possible outcomes in the sample space, as shown in Table 3-2. The outcomes are denoted as ordered pairs, with the face-number of the red die listed first and the face-number of the blue die listed second.

Mathematical Probability

Let x be a discrete random variable that can attain n possible values, all equally likely. Suppose an outcome H results from exactly m different values of x, where m ≤ n. Then the mathematical probability p_math(H) that outcome H will result from any given value of x is given by the following formula:

p_math(H) = m/n

Expressed as a percentage, the probability p%(H) is:

p_math%(H) = 100m/n

If we toss an ''unweighted'' die once, each of the six faces is as likely to turn up as each of the others. That is, we are as likely to see 1 as we are to see 2, 3, 4, 5, or 6. In this case, there are six possible values, so n = 6. The mathematical probability of any particular face turning up (m = 1) is equal to p_math(H) = 1/6. To calculate the mathematical probability of either of any two different faces turning up (say 3 or 5), we set m = 2; therefore p_math(H) = 2/6 =1/3. If we want to know the mathematical probability that any one of the six faces will turn up, we set m = 6, so the formula gives us p_math(H) = 6/6 = 1. The respective percentages p_math%(H) in these cases are 16.67% (approximately), 33.33% (approximately), and 100% (exactly).

Mathematical probabilities can only exist within the range 0 to 1 (or 0% to 100%) inclusive. The following formulas describe this constraint:

    0 ≤ p_math(H) ≤ 1

0% ≤ p_math%(H) ≤ 100%

We can never have a mathematical probability of 2, or –45%, or –6, or 556%. When you give this some thought, it is obvious. There is no way for something to happen less often than never. It's also impossible for something to happen more often than all the time.