Mutually Exclusive Probability

Mutually Exclusive Probability

How do we calculate mutually exclusive probability? A useful vehicle for explaining this calculation is a jar containing 90 jelly beans: 45 red ones, 36 yellow, and 9 green. We’re going to reach into the jar and pick out one bean. We can visualize this more clearly with a scenario tree (Figure 13-1) in which we have three possible, mutually exclusive decisions or outcomes: red, yellow, and green.

FIGURE 13-1

Assuming each bean has an equal chance of being selected, what is the probability, if I reach into the jar and blindly take out a single jelly bean, that it will be red? What probability yellow? What probability green?

One easy way to compute mutually exclusive probabilities is to think of them as percentages. For example, the probability of picking one red bean out of 90 in one try is equal to the percentage of the 90 that are red. Does that make sense? Maybe it’s clearer if I rephrase the sentence. If, in a different jar, 30 percent of the beans are red, then I have a 30 percent chance of picking a red bean on one try.

How, then, do we compute the percentage of red beans in the jar? I like to use the traditional IS-over-OF method: IS divided by OF. I learned it in grade school and have never found a better way to structure the arithmetic formula. (Yes, IS-over-OF is a structuring technique.) To do the calculation, we ask ourselves, “What percent is this number of that one?” We then divide the IS number by the OF number. For instance, what percent is 3 of 6? Or what percent of 6 is 3? IS (3) over OF (6): 3 divided by 6 equals .5 (50 percent).

One advantage in using the IS-over-OF formulation is that sometimes the context of a problem makes it difficult to determine which number is to be divided by which. I was taught, when encountering this dilemma, to transform the language of the problem into an IS-over-OF format. To wit, a bushel contains 40 apples, 10 of which are rotten. What percentage are rotten? Some people might be confused as to which number is the divisor and which the dividend. To find out, one simply rewords the question as follows: What percent is 10 of 40? IS-over-OF; 10 over 40 = .25 (25 percent). I love it! (God bless my grade school teachers.)