Introduction to probability
As you have most likely learned in the past, probability is the ratio of the number of ways in which a successful outcome can occur to the total number of possible outcomes. Counting the number of ways in which success can occur can be tricky. This chapter begins with counting issues, and works toward computing probabilities of special cases.
Fundamental theorem of counting
A classic example of counting is the multistep problem. The fundamental theorem of counting states that the total number of ways in which a task can be completed is equal to the product of the number of ways in which each step in the sequence can be completed. Two examples are ordering a meal from a menu and creating a personal identification number (PIN).
The product of the first n counting numbers is called n factorial and is written as n!. The product of the first five counting numbers is 5! = 120, as was seen in the last example.
Solve the following problems.
1. A restaurant is offering a special deal for a three-course meal at a fixed price. The meal consists of a salad, an entrée, and a dessert. If the menu contains two salad choices, five entrées, and three desserts, how many different meals can be ordered?
2. “I scream! You scream! We all scream for ice cream!” Patty goes to her favorite ice cream parlor to order a triple-scoop cone with sprinkles. When she gets there, she is pleased to see that the parlor offers 12 different flavors of ice cream. She has a choice of a sugar, wafer, or waffle cone, and they have chocolate as well as rainbow sprinkles. How many different ice cream cones can Patty order?
3. Compute 6! + 4!
4. Compute (3!)2 + (32)!
5. Compute 
6. Compute 
7. Alan, Bob, Colin, Don, and Ed had such a good time taking their pictures, they talk their friends Frank and George into joining them the next day to have pictures of the seven of them taken together. With 15 sec between pictures, how much time will it take for the seven of them to have their group pictures taken in all possible orders?
Permutations
There are times when the order of arrangement of objects is important, and other times when it is not. A permutation is an arrangement of objects in which order matters. A combination is an arrangement of objects in which order does not matter.
Order matters when the arrangement represents a code. Words are codes (BEAT and ABET use the same letters but are different words). License plate tags, telephone numbers, PINs, and security access codes are all examples of permutations. (Note: The access code to get into your locker really is your locker permutation, not your locker combination. This is an unfortunate mix-up in the use of the words.)
The two examples hint at the formula for the number of permutations of r objects taken from a group of n objects: 
In general, if a word with n letters has r1 repetitions of one letter, r2 repetitions of a second letter, etc., the number of arrangements of the letters is 
Solve the following.
1. Compute 9P5.
2. Compute 10P6.
3. How many different arrangements can be made using all of the letters in the word KATHRYN?
4. How many different arrangements can be made using all of the letters in the word RUSSELL?
5. The locks on the lockers in the high school have a three-digit code chosen from the 36 numbers on the dial. If the code consists of three different numbers, how many different codes can be made?
6. The positions on a basketball team are assigned numbers to designate their responsibilities (for example, the #1 player is the point guard while #2 is the shooting guard). There are five different positions during a scrimmage, and there are 12 players on the team. If Coach Treanor randomly assigns five of these players to different positions, how many different teams can Coach Treanor put on the floor?
7. Stacey is one of the 12 players on Coach Treanor’s team. What is the probability that she is on the team chosen to play?
8. Kieran’s Little League team has 15 players on the roster. He puts each name on a card and puts the cards into a hat. His batting order for the day will be the first nine players he picks, with the players batting in the order in which they are picked. What is the probability that Will and Peter, two of his players, will be the first two players to bat?
Combinations
Alice, Barbara, Cathy, Donna, and Edith want to have their pictures taken three at a time. They do not care about who stands next to whom, or the order in which they are standing. How many different pictures need to be taken? You have seen that there were 60 different pictures needed when the guys had their pictures done. How does this number change if order is not a consideration? Suppose Alice, Barbara, and Cathy are the first three to have their pictures taken. If order mattered, ABC, ACB, BAC, BCA, CAB, and CBA (using the first letters of their names) would all be different arrangements. Since order does not count, these 3!(6) arrangements are all the same. Therefore, according to the girls’ requirements, there are only 
pictures that need to be taken.
The number of combinations of n things taken r at a time is 
Another notation for 
.
Pascal’s triangle, an interesting item in the study of mathematics, provides a graphical approach for computing combinations. The triangle can be constructed from two different algorithms. The first involves a recursion formula. Each row begins and ends in a
With the top row numbered n = 0, the second row numbered n = 1, etc., each entry in the triangle can be computed using the combination formula 
where r starts with 0 and finishes with n. For example, the last line in the diagram shown is n = 5. Thus, the numbers for this line are computed as
Compute the number of possible combinations for the following.
1. Compute 10C6.
2. Compute 12C8.
Seven students are to be selected to work on the homecoming committee. There are 12 seniors and 10 juniors to choose from. Given this information, answer the following.
3. How many different possible combinations are there for the committee?
4. What is the probability that the committee formed will contain more seniors than juniors?
5. Marian and Kristen are two of the eligible students. What is the probability that they will both be on the committee?
Eight cards are placed in a box. On each card is written the lengths of the sides of a triangle (3, 4, 5; 5, 12, 13; 1, 
2; 8, 15, 19; 11, 60, 61; 7, 24, 25; 20, 30, 40; 12, 16, 21). Given this information, solve the following.
6. If four cards are drawn at random from the box, what is the probability that three of them will have lengths that represent a right triangle?
Binomial expansions
The expansions for (a + b)n are shown for n = 0, 1, 2, 3, and 4.
The following observations can be made from these expansions:
The number of terms in the expansion of (a + b)n is n + 1.
The exponents on a begin with n and decrease by 1 to reach 0 (remember a0 = 1).
The exponents on b begin with 0 and increase by 1 to reach n.
The coefficients for each of the expansions are the same as seen in Pascal’s triangle.
Each coefficient for row n in Pascal’s triangle is 
where r begins with 0 and grows to n (the same as the exponents for b).
Expand the two following expressions.
1. (2x + y)5
2. (x − 2y)6
Find the specified term in the expansion of the three binomials given below.
3. The fifth term in the expansion of (3a − 4b)9
4. The middle term in the expansion of 
5. The last term of the expansion of 
Find the coefficient of x4 in the expansion of the following binomial.
6. (2x − 3)9
Conditional probability
What is the probability of selecting an 8 from a well-shuffled bridge deck (52 cards in four suits—spades, hearts, clubs, and diamonds)? Since there is one 8 in each of the four suits, the probability of getting an 8 is 
This type of question places no conditions on the likelihood of achieving the intended outcome. A problem of this type is known as an unconditional probability problem.
Using the same well-shuffled deck, what is the probability of selecting an 8 on the second draw, if the first card drawn (and not returned to the deck) was a king? The difference between this question and the one in the previous paragraph is that there is a condition attached: the first card drawn, and not replaced, is a king. The probability of getting an 8 on the second pick, given that a king is drawn on the first pick, is written P(8|K), and is 
This is an example of conditional probability.
Two events, A and B, are said to be independent of each other, if P(B|A) = P(B). That is, the first outcome has no impact on the opportunity for the second outcome to occur. A consequence of this is that two events are independent if and only if P(A and B) = P(A) × P(B).
Solve the following.
1. What is the probability that the second card drawn from a standard bridge deck of cards is a spade given that the first card drawn (and not returned to the deck) is a heart?
Use the accompanying Venn diagram, which represents the members of the senior class at a small high school, to answer questions 2–4.
2. If a member of the senior is selected at random, what is the probability that the student is a member of the honor society given that the student is in orchestra?
3. If a member of the senior is selected at random, what is the probability that the student is not a member of the orchestra? Given that the student is in the honor society?
4. Are the events “a senior is a member of the honor society” and “a senior is a member of the orchestra” independent of each other?
Use the accompanying table, which represents the amount of time a person listens to music each day, to answer questions 5–7.
5. What is the probability that a randomly selected person from this survey listens to music 1–2 hr each day, if that person is between the ages of 15 and 17?
6. What is the probability that a randomly selected person from this survey is between the ages of 18 and 20, if that person listens to music for 4 or more hours each day?
7. Are the events “the person is between 21 and 24 years old” and “the person listens to music 2–3 hr each day” independent of each other?
Binomial probability/Bernoulli trial
A probability experiment with the properties that
there are n independent trials;
there are two outcomes per trial (success and failure); and,
P(success) is constant from trial to trial
is called a binomial experiment. (It is also called a Bernoulli trial after the family of Swiss mathematicians.) The probability of exactly r successes in n trials is given by the formula 
where p represents the probability of success on any given trial.
This is key—it is not important if the roll is a two or five, whether or not it is one is the only thing that matters.
There is a function on most graphing calculators called binomCdf that will compute the probability of a range of events. Look at the directions for your graphing device to see if you have it.
Use the diagram of the spinner with six congruent regions to answer questions 1–3.
1. If the spinner is spun six times, what is the probability of the result being red three times?
2. If the spinner is spun five times, what is the probability of the result being blue at most twice?
3. If the spinner is spun 10 times, what is the probability of the result being green at least twice?
Based on past statistics, a company knows that 99% of its ball bearings pass a quality control test. A random sample of 100 ball bearings is tested for the quality control test.
4. What is the probability that all 100 bearings will pass the test?
Laura is one of the school’s best tennis players. Laura’s rate of winning first-serve points is 78%.
5. What is the probability that Laura will win at most eight first-serve points in her next 10 serves?