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**

1.
Lollipops

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.)

## No comments:

## Post a Comment