Suppose four fair coins are tossed at the same time that these two spinners are spun. Using the fundamental counting principle, find the total number of possible outcomes.
The fundamental counting principle is a method we can use to find the number of all possible outcomes in a sample space. This tells us that for two independent events 𝐴 and 𝐵 such that the number of possible outcomes for event 𝐴 is 𝑥 and the number of possible outcomes for event 𝐵 is 𝑦, the total number of distinct possible outcomes for the two events together is the product 𝑥 multiplied by 𝑦. This can be extended to any number of independent events. To find the total number of outcomes, we find the product of the number of outcomes for each individual event.
In this question, we have four fair coins and then we have the two spinners shown. Coins have two faces, so the number of possible outcomes for each of the four coins is two. For the first spinner, we see that it has four quadrants each in a different color. So there are four possible outcomes for the first spinner: blue, green, yellow, or red. The second spinner has sectors in different colors, but it also has different letters, each of the letters from 𝐴 to 𝐻. So there are eight possible outcomes for the second spinner.
The fundamental counting principle tells us to find the total number of possible outcomes for the four coins and two spinners, we multiply all of the individual numbers of outcomes together. So we have two times two times two times two times four times eight. Now, this is two to the power of four or two to the fourth power for the four coins multiplied by two squared for the first spinner multiplied by two cubed for the second spinner, which is equal to two to the power of nine. And finally, evaluating two to the ninth power perhaps by repeatedly doubling two, we see that this is equal to 512.
So by the fundamental counting principle, we found the total number of possible outcomes when we toss four coins and spin these two spinners is 512.