Page 284 - ABCTE Study Guide_Neat

Choice C is the correct answer. Of the numbers 1 through 6, three are even. So the probability of rolling

an even number is , or 50%.

Counting Methods

A sample space is the set of possible outcomes for an event. For example, the sample space for flipping a
coin is {heads, tails}. The sample space for rolling a six-sided die is {1, 2, 3, 4, 5, 6}.

Because probability is the ratio of the number of desired outcomes to the total number of outcomes, when
sample spaces get very large, techniques for counting the number of elements in a sample space are
extremely useful.

For example, how many ways are there to line up ten people in a row? Actually, there are more than three
million ways to do it. We definitely need an organized and systematic way to figure this out. Enter the
world of permutations and probability.

Combinations and Permutations

Combinations

In more everyday terms, a combination is an un-ordered selection from a group of objects. For example,
let's say you have fifty-two cards and select five random cards - for a hand of poker. It does not matter in
what order the cards are drawn, because you can rearrange them without a loss of information. This is the
crucial difference between combinations and permutations.

In more formal mathematical terms, a combination is a subset of a set. In a set of objects, the order of the
objects does not matter. And since order does not matter, we are only interested in what objects are
present, not their order. So, in a combination {2, 4, 6} = {6, 4, 2} = {4, 2, 6}.

Permutations

On the other hand a permutation is a specifically ordered selection made from a group of objects. Lets
use the card example again. This time, however, the most important aspect of our subset is its order or
arrangement of objects. For example, if we drew a 5 of clubs, a J of diamonds, a 7 of spades, an 8 of
clubs, and a 10 of clubs - this is NOT the same as that assemblage in a different order: a J of diamonds,
an 8 of clubs, a 5 of clubs, a 7 of spades, and a 10 of clubs.

Here is an example with a set of three objects. There are six permutations of a red (R), a green (G), and a
blue (B) marble:

RGB, RBG, GBR, GRB, BRG, BGR

Notice that the order matters. Every arrangement consists of all three marbles, but each gives the marbles
in a different order.

As the number of items increases, the number of permutations of those items increases incredibly faster.
Hence, when you hear that there are more than 5,000 ways to prepare a burger, know that it takes only
seven toppings to do that.

Suppose these are the burger toppings we’ve been talking about: