Combinatorics is an area of mathematics that deals with the study of combination, enumeration, and permutations of sets of elements. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry. There are many subfields of combinatorics, but we will study about special subfields of combinatorics, that is probability.
Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes, relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%.
Probability measures the likelihood of a specified event occurring. Probability can present itself as a ratio, or fraction, where the numerator is the number of different ways the event could occur and the denominator is the total of all possible outcomes. We will introduce probability in several ways.
a. Introducing Probability Using Cards
A standard deck of playing cards can be used to show how proportions work because the basic concept of a deck of cards is familiar to most students. The deck has 52 cards total.
1. Divide the deck into suits (heart, spades, diamonds and clubs) and show the students that each suit contains 13 cards. Point out that each suit contains three face cards (jack, king and queen) and since there are four suits, there are 12 (3 * 4) total face cards in the deck. Shuffle the deck of cards to mix them back up.
2. Ask the students what the probability is of you drawing a two of hearts from the shuffled deck. Ask how many cards there are, total, in the deck. Write 52 at the bottom of a fraction. Ask how many chances you have of pulling out the two of hearts specifically. Write 1 in the numerator for an answer of 1/52.
3. Ask the students what the probability is that you would draw a spade from the deck. Ask what the numerator is then write 13 on the board, explaining that's the total number of cards in the suit of spades. Write 52 as the denominator and simplify the fraction to (1/4).
4. Ask what the probability is that you would draw a two from the deck. Question the students on how many two cards are in a deck: There are four, or one for each suit. Write the probability as 4 / 52, which simplifies to 1/13.
b. Introducing Probability Using Games
The lollipop game is a simple but fun way to experiment with probability. A simple probability game for two students to construct is to take a large rectangle of Styrofoam and stick lollipops into it. The lollipops can be arranged in 10 rows of 10. Mark each lollipop stick near its end with a bright marker. There should be different quantities of each color. The lollipops should be pushed far enough into the Styrofoam to hide the colored markings. Students will determine the probability of picking out one of several possible colors. Students can choose the number of different colors there should be and how many there should be of each color. Students should calculate the probability of the different possible outcomes with their partner. For example, if there are 100 lollipops, and 20 of them are red, then the probability of drawing a red is 2 in 10, or 20 percent.
2. Roll the Dice
This game might be a bit more fun with oversized dice, but can be played with traditional small dice as well. To make it challenging for middle school students, have them calculate the probability of not only total sums of the dice i.e., "rolling a 10," but of particular combinations, as in "rolling two fives." To make the game even more challenging, three or more dice can be used. Students can create a fun, decorated "arena" to roll the dice in, using the lid of a dress box or something similar. Students should calculate the possibility of the possible outcomes, for example, if there are two dice, the probability of rolling two threes is 1 in 36, or almost 3 percent. (Of the 36 possible outcomes when rolling a pair of dice, only one of those outcomes is double threes.)