Consider also the analysis of Projects A and B. Suppose Project A has a potential return of $100,000, while Project B yields only $80,000. As A and B are mutually exclusive, the opportunity costs of choosing B are the gain of the most lucrative option (in this case A) minus the gains generated by the selected option (B); That is, $100,000 – $80,000 – $20,000. As Option A is the most lucrative option, the opportunity cost of Option A is $0. In statistical and regression analysis, an independent variable, which can take only two possible values, is called “dummy” For example, it can take the value 0 if an observation is of a white subject or 1, if the observation is of a black subject. The two possible categories, linked to the two possible values, are mutually exclusive, so that no observations fall under more than one category and the categories are exhaustive, so that each observation falls into a category. There are sometimes three or more possible categories that are mutually exclusive in pairs and that are collectively exhaustive, for example. B under the age of 18, 18 to 64 and 65 or older. In this case, a set of model variables is constructed, each model variable containing two mutually exclusive and collectively exhaustive categories – in this example, a dummy variable (d1) would correspond to 1 if the age is less than 18 years, and if not, it would correspond to 0; A second model variable (d2) would be 1 if the age is 18-64, and 0 if not. In this device, pairs of model variables (D1, D2) have values (1.0) (18), (0.1) (between 18 and 64) or (0.0) (65 or more) (but not (1.1), implying insensitively that an observed subject is both under 18 and between 18 and 64 years of age. Dummy variables can then be included in a regression as independent (explanatory) variables. Note that the number of model variables is always smaller than the number of categories: with the two black and white categories, there is only one model variable to distinguish them, while for the three age categories, two variables of models are needed to distinguish them. For example, in a standard deck of 52 cards with two colors, it is impossible to draw a card that is both red and club, because the clubs are always black.
If only one card is removed from the deck, a red card (heart or diamond) or a black card (club or spat) is drawn. If A and B are mutually exclusive, P (A ∪ B) – P (A) – P (B) are mutually exclusive.  For example, to determine the probability of shooting a red card or bat, add the probability of shooting a red card and the probability of drawing a bat. In a standard deck of 52 cards, there are 26 red cards and 13 rackets: 26/52 – 13/52 – 39/52 or 3/4. Events are collectively exhaustive when all possibilities for the outcome of these possible events are exhausted, so that at least one of these results must occur. The probability of at least one of the events occurring is equal to one.  For example, in theory, there are only two ways to flip a coin.