subject: Counting Principle Probability [print this page] Counting Principle Probability Counting Principle Probability
Counting possibilities is the process of counting the number of possible ways to do thing, or number of samples or number of outcomes. Counting possibilities is necessary to handle large statistical data and solve the probability problems.
Counting Principle:
If there are m number of possible ways to do one thing, and n number of possible ways to do another thing, then there are m x n number of possible ways of doing both the things.
Counting Principle Probability - Example Problems
Example 1: To buy 4 different kinds of marbles, a customer can choose 5 of 10 red marbles, 4 of 8 orange marbles. Find the number of possible ways customer can choose from.
Solution:
We have to choose 5 red marbles out of 10 and 4 orange marbles out of 8.
Number of possible ways = C(10, 5) x C(8, 4) = 252 x 70 = 17640
Example 2: There are 10 dark and 8 bitter chocolates in a jar. In how many ways can draw 6 chocolates, with 2 bitter chocolates from a jar?
Solution:
We have to choose 4 dark chocolates out of 10 and 2 bitter chocolates out of 8.
= C(10, 4) x C(8, 2) = 210 x 28 = 5880
Therefore, 5880 ways can draw 6 chocolates out of 18 chocolates.
Example 3: In a class there are 20 boys and 20 girls. In how many ways can make a group with 2 boys and 2 girls?
Solution:
Boys: C(20, 2) = 190
Girls: C(20, 2) = 190
C(20, 2) x C(20, 2) = 190 x 190 = 36100
In 36100 ways can make a group.
Counting Principle Probability - Practice Problems
Solve these practice problems using counting principle probability.
Problem 1: There are 8 dark and 7 bitter chocolates in a jar. In how many ways can draw 4 chocolates, with 2 bitter chocolates from a jar?
Problem 2: In a class there are 24 boys and 15 girls. In how many ways can make a group with 3 boys and 3 girls?
Answer: 1) 588 2) 920920
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Where really need the Fundamental Counting Principle in high school and early college mathematics is in the study of probability. In particular, when we want to dig down deep into a sample space, we often need to use the Fundamental Counting Principle to make sure we have identified all possible outcomes.
One of the biggest mistakes in all of probability is to assume that the elements of your sample space are equally likely. Finding theoretical probabilities is much easier if we have a sample space composed of equally likely elements, so this is a desirable thing to have, but it does not follow that listing out the sample space means the elements are equally likely.
So, we use the Counting Principle to determine the different unique ways we can do something, such as a sandwich or a selection of classes. Sometimes, these events will affect each other, such as when you can't choose the same number twice for your garage door code, so they are dependent events. However, other times, one event has no effect on the next event, such as when you have different cheeses and breads to choose for your sandwich, so they are independent events. The Counting Principle is a fundamental mathematical idea and an essential part of probability.
