Lesson 1 Part 1: Counting Independent Events

Starting off...

Lesson 1: Counting

Part 1: Counting Independent Events

Let's say we were trying to solve the following problem:

There are 5 pickles, 9 flags, 2 laptop cases, and 11 Christmas trees. How many ways can you choose one pickle, one flag, one laptop case, and one Christmas tree.

How would you do this problem?

Well, there are 5 ways to choose a pickle, 9 ways to choose a flag, 2 ways to choose a laptop case, and 11 ways to choose a Christmas tree. Let's pick a pickle first. There 5 possibilities. Now, let's choose a flag. There are 9 possibilities for flag choices for each of the 5 pickles we could have chosen, so there is 5x9=45 ways to choose a pickle and a flag. The same logic applies to each of these pickle-and-flag combinations, with there being 2 ways to choose a laptop case for each of these (45x2=90 combinations in all), and then the Christmas tree, with 11 ways to choose a Christmas tree for each of those 90 possible combinations of the other three. So, there are 90x11=990 ways to choose a pickle, a flag, a laptop and a Christmas tree.

Well, that was painfully obvious, that would would multiply the number of possibilities for each to come up with the answer. However, this is important in the sense that it is the BASE for ALL counting problems. Since the factors or "events" are completely independent of each other (one does not change the other in some way), we call this counting independent events. When counting independent events, we are not concerned with the factors in itself, but with the number of possibilities each factor gives (note that the number might remain the same while the factors chance!). If you want a prime example, try do #*3 in the problems below.

You can also show counting independent events with a probability tree, with the first event branching out in each of its possibilities, and then the second event coming out from each branch.

Problems:
Counting independent Events

1. Emile wants to eat ice cream. The ice cream shop offers 5 flavors of ice cream, and 3 scoop sizes. He can choose to eat it on a sugar cone, a plain cone, or without a cone. Finally, the shop has 3 flavors of sprinkles in 4 different shapes each. How many possible different ways can Emile order his ice cream?

Counting independent probabilities
2. Emile wants to play the flute. There is an equal chance that he will rent his flute from the school or he will buy his flute. There is a 9/17 chance that he will get a Pearl flute. There is a 1/99 chance that he will get a silver, open-holed flute. Out of all silver open-holed Pearl flutes, 1/3 is the exact variety Sean plays. There is a 1/8 chance that there will be a random sentence in the problem to throw you off. What are the odds he the exact same kind of flute as Sean?

Trickier counting, by a bit.

*3. A painter named Emile wants to paint his long 1x9 grid wall with a different color on each grid, under the condition that no two adjacent grids be painted the same color. The painter has 4 colors of paint to choose from. How many ways can he paint his wall?