Decision algorithm
A
decision algorithm is a systematic list of decisions we need to make in order to completely determine an outcome of some activity. We can use a decision algorithm to determine the number of possible outcomes by applying the addition and multiplcation principles to the decision algorithm.
In short:
*What it means for a decision algorithm to be
valid is that making different choices must always result in a different outcome. For instance, if you look at the decision algorithm we wrote down to decide on a gift pen set, making different decisions in any of the three steps would result in a different gift.
#[Example][Ejemplo]#
You are the coach of the Enormous State U Blastball team. Your team is to have four players: a striker, two blasters, and a blocker. You have five possible players to choose from to make up the team.
Following are two decision algorithms; one are valid and one is invalid. In the first algorithm, you pick players for the five positions on-by-one to form the team. In the second, you instead fill the three positions one-by-one:
\t
Decision algorithm 1
Step 1: Pick the striker: 5 choices
Step 2: Pick the first blaster: 4 choices
Step 3: Pick the second blaster: 3 choices
Step 4: Pick the blocker: 2 choices
Total: 5×4×3×2 = 120 possible teams
\t
Decision algorithm 2
Step 1: Pick the striker: 5 choices
Step 2: Pick the blocker: 4 choices
Step 3: Pick two of the remaining three to be the blasters: 3 choices
Total: 5×4×3 = 60 possible teams
Although each of these algorithms leads to a blastball team, Algorithm 1 is
not valid for counting the number of possible teams you can make. Why? Suppose that Joaquim and Matilda are two players. Then, if you choose Joaquim in Step 2 and Matilda in Step 3, you end up wit the same two blasters as you would by choosing Matilda in Step 2 and Joaquim in Step 3. Thus switching your decisions in Steps 2 and 3 results in the same outcome (the same team), and so the Algorithm 1 is not valid.
On the other hand, if you make changes any of your decisions in Algorithm 2, the resulting team will be different (think about why). Tus, Algorithm 2 is valid.