target="_blank" rel="nofollow" href="#fb3_img_img_2a728ef5-14c4-5384-a609-37ecedf487c2.png" alt="upper P left-parenthesis omega right-parenthesis equals 1 slash k"/>, for all Computing probabilities for equally likely outcomes takes a fairly simple form. Suppose
In other words, probability with equally likely outcomes reduces to counting elements in
Example 1.9 A palindrome is a word that reads the same forward or backward. Examples include mom, civic, and rotator. Pick a three-letter “word” at random choosing from D, O, or G for each letter. What is the probability that the resulting word is a palindrome? (Words in this context do not need to be real words in English, e.g., is a palindrome.)There are 27 possible words (three possibilities for each of the three letters). List and count the palindromes: DDD, OOO, GGG, DOD, DGD, ODO, OGO, GDG, and GOG. The probability of getting a palindrome is
Example 1.10 A bowl has red balls and blue balls. A ball is drawn randomly from the bowl. What is the probability of selecting a red ball?The sample space consists of balls. The event has elements. Therefore, .
A model for equally likely outcomes assumes a finite sample space. Interestingly, it is impossible to have a probability model of equally likely outcomes on an infinite sample space. To see why, suppose
While equally likely outcomes are not possible in the infinite case, there are many ways to assign probabilities for an infinite sample space where outcomes are not equally likely. For instance, let
We introduce some basic counting principles in the next two sections because counting plays a fundamental role in probability when outcomes are equally likely.
1.6 COUNTING I
Counting sets is sometimes not as easy as
Multiplication principle
If there are
More generally—and more formally—consider an
For instance, in tossing a coin three times, there are
Example 1.11 License plates in Minnesota are issued with three letters from A to Z followed by three digits from 0 to 9. If each license plate is equally likely, what is the probability that a random license plate starts with G-Z-N?The solution will be equal to the number of license plates that start with G-Z-N divided by the total number of license plates. By the multiplication principle, there are possible license plates.For the number of plates that start with
