AFM Guided NotesUnit 5 – Probability

5.1 Combinations & Permutations

Counting Principals Have you ever imagined what your life would be like if you won the lottery? What changes would you make? Before you fantasize about becoming a person of leisure with a staff, think about this: The probability of winning the top prize in the lottery is about the same as the probability of being struck my lighting. In this section, we begin preparing you for the surprising world of probability by looking at methods for counting possible outcomes.

_________________________________ is a way to figure out the total number of ways different events can occur. For example: ________________ outfits or how many ______________ sandwiches you can make.

m =

n =

p =

Formula:

If you have _______________ _______ events, ______________ the number of times each event occurs.

Examples: 1. Suppose you own a small deli. You offer 4 types of meat (ham, turkey, roast beef and pastrami) and 3 types of bread (white, wheat, and rye). How many choices do your customers have for a meat sandwich?

2. 2. You are going on vacation. You pack one pair of khakis, one pair of jeans, one blue t-shirt and one green t-shirt, one striped button-down shirt, an pair of tennis shoes, and a pair of sandals. How many outfits can you make?

3. At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12 different drinks, and 6 different desserts. How many different dinners consisting of 1 of each, can you choose?

4. Police use photographs of various facial features to help witnesses identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheeks. Can this kit produce billions of faces as the develop claims?

5. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheeks. If a witness clearly remembers the hairline and eyes and eyebrows of a suspect, how many different faces can be produced?

6. It’s early morning, you’re groggy, and you have to select something to wear for school. You have two pairs of jeans (blue and black), four t-shirts (yellow, red, white, and blue), and two pairs of sneakers (black and green). Find the possible number of outfits that you can make.

Permutations_____________________________ is a way in which a set or number of things can be _________________ or _________________. __________________________!

There are basically two types of permutations:

Permutations with repetition: These are the easiest to calculate.

When you have _____ things to choose from…. you have _____ choices each time.

When choosing ____ of them, the permutations: ___________________________ = ____________

1. Using the digits 4,5,6, and 7, how many two-digit numbers can be formed with repetition?

2. Using the digits 4,5,6,7,8, and 9, how many five-digit numbers can be formed with repetition?

3. Let’s say that the new license plates only contain the alphabet. There are still 7 spots for each letter. How many license plates are formed if you can repeat the letters?

Permutations without repetition

Formula = n =

r =

Examples: 1. Ten teams are competing in the final round of the Olympic four-person bobsledding competition. In how many different ways can 3 bobsledding teams finish first, second, and third to win the gold, silver, and bronze medal?

2. You are burning a demo CD for your band. Your band has 12 songs stored on your computer. However, you want to put only 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD?

3. You are considering 10 different colleges. Before you decide to apply to the colleges, you want to visit some or all of them. In how many orders can you visit:

a) 6 of the colleges?

b) all 10 colleges?

4. How many 4 letter words can be created if repetitions are not allowed?

5. A baseball league has 13 teams, and each team plays each other twice; once at home, and once away. How many games are scheduled?

Combinations____________________________ is a way in which a set or number of things can be ordered or arranged. Order __________________ matter!

Formula: n = r =

1. The volleyball team has 9 players, but only 6 can be on the court at one time. How many different ways can the team fill the court?

2. A painter was carrying 7 pails of different colored pain and dropped 4 of them, making a big mess. How many combinations of colors could he have spilled?

3. Natalie asked the pet store owner for any 5 baby mice from a cage containing 9. How many possible combinations of mice could be picked?

4. 5 names will be picked from a jar to be on a team. There are a total of 11 names in the jar. How many different combinations of names can be picked?

Difference between permutations and combinations: Permutation:

Combination:

Is this a combination or permutation? 1. There are 8 people competing in a race. How many different ways can first, second, and

third place medals be awarded?

2. A pizza shop offers twelve different toppings. How many different three-topping pizzas can be formed with the twelve toppings?

3. Your English teacher has asked you to select 3 novels from a list of 10 to read as an independent project. In how many ways can you choose which books to read?

4. The school yearbook has an editor-in-chief and an assistant editor-in-chief. The staff has 15 students. How many different ways can students be chosen for these 2 positions?

5.2 Probability

• How ___________________ something is to happen. • Many events can’t be predicted with total certainty. The best we can do is say how

____________they are to happen, using the idea of ____________________.

You can express probabilities in three different ways:

Probability ranges from . All probabilities to 1 or 100.

When a _________ is tossed, there are _______ possible outcomes.

Experimental vs. Theoretical

Experimental Theoretical

Given the following situations, determine the type of probability. 1. A bag contains three red marbles and three blue marbles. What is the probability of

choosing a red marble? 2. You draw a marble out of the bag, record the color, and replace the marble. After 6 draws,

you record 2 red marbles.

Theoretical Probability: =

An is the result of a single trial of an experiment.

When flipping a coin, you have two outcomes:

When rolling a die, you have six outcomes:

A is the set of all possible outcomes.

Example: Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl.

Example: 1. What are the chances of rolling a “4” with a die?

2. There are 5 marbles in a bag: 4 are blue and 1 is red. What is the probability that a blue marble will be picked? 3. Of the 60 vehicles in a teacher’s parking lot today, there are 15 pickup trucks. What is the probability that a vehicle in the lot is a pickup truck?

4. What is the probability of each event?a) getting a 5 on one roll of a die.

b) getting a sum of 5 on one roll of two die.

5. A bucket contains 15 blue pens, 35 black pens, and 40 red pens. You pick one pen at random. Find the probability a) P(black pen) b) P(not a blue pen) c) P(blue pen)

d) P(not a blue pen)

Probability = 1 or 100 If you have two events A and B that can happen, then

If you have three events A, B, & C that can happen, then

So, the probability of all events happening must add up to .

Probability an event does not occur: If you know the probability than an event , then:

P(not A) =

Example: Example: The probability that Billy’s birthday is today is 0.0027. What is the probability that it is not his birthday today?

Addition Rules for ProbabilityMany problems involve finding the probability of events.

Two events are events if they cannot occur at the same time (they have no outcomes in common).

Example: With playing cards, the events of getting a 4 and getting a 6 when a single card is drawn from a deck are events, since a single card cannot be both a 4 and a 6.

Not Mutually Exclusive: The events of getting a 4 and getting a heart on a single draw are

mutually exclusive, since you can select the when drawing a single card.

The events of rolling a die and getting a number less than 4 and an even number are not mutually exclusive, since you can roll a 2, .

Are the following Exclusive? When a single die is rolled:

1. Getting an odd number and getting an even number.

2. Getting a 3 and an odd number.

3. Getting an odd number and getting a number less than 4.

4. Getting a number greater than 4 and getting a number less than 4.

If the events are Mutually Exclusive: Addition Rule #1:

Examples: 1. A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random, find the probability that it is either a glazed doughnut or a chocolate doughnut.

2. At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is ether a Democrat or an Independent?

3. A day of the week is selected at random. Find the probability that it is a weekend day.

If the events are Mutually Exclusive: Addition Rule #2:

1. A single card is drawn from an ordinary deck of cards. Find the probability that it is either an ace or a black card.

2. In a hospital unit there are 8 nurses and 5 physicians, 7 nurses and three physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

3. On New Year’s Eve, the probability of a person driving while intoxicated is 0.32, the probability of a person having a driving accident is 0.09, and the probability of a person having a driving accident while intoxicated is 0.06. What is the probability of a person driving while intoxicated or having a driving accident?

Independent Events: Events that are if the occurrence of one event the probability of the other.

For example: If a coin is tossed twice, the of one toss does not affect the of the second toss.

Another example: Drawing a queen from a deck of cards, replacing it, then drawing a queen again.

Tree Diagrams

are very helpful when thinking of independent events.

This is a tree diagram that represents tossing a coin once. • The of each branch is • The is written at the of the branch.

We can extend the tree diagram to compute the probability of tossing two coins.

You can calculate overall probabilities by: along the branches.

All probabilities should add to , so this is a way to check that you did it correctly.

Independent EventsIf the events are independent, (meaning the outcome of one does not depend on the other), then if you see the word , you can the branches.

Multiplication Rule:

Examples: 1. After tossing the coin twice, what is the probability that it will land Heads both times?

2. If you flip a coin and roll a die, what is the probability of getting a Heads and a 4?

3. If you spin the color spinner and the number spinner, what is the probability of landing on red and a 4?

4. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.

5. A card is drawn from a deck and replaced, then a second card is drawn. Find the probability of getting a queen and then an ace.

6. A Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three wills say that they suffer great stress at least once a week.

7. If the probability of giving birth to a boy is 0.52, what is the probability of giving birth to four consecutive boys?

8. At a local university 54.3% of incoming first-year students have computers. If 3 students are selected at random, find the following probabilities.

a) None have computers

b) All have computers.

Playing Card Cheat Sheet:

5.3 Binomial Probability

There are many probability experiments for which the results of each trial can be reduced to outcomes:

ex: when a basketball player attempts a free throw. What can happen?

A has these important features: There are a number of trials Each trial has possible outcomes

The trials are

the outcome of one trial the probability of success on any other trial.

The probability of each outcome is throughout the trials

There are two outcomes for a binomial experiment:

NotationSymbol Description

Are you right or left handed? 1. In a tennis league, 80% of the players are right-handed. The league president is randomly selecting seven players to demonstrate serves. What is the probability that exactly three of the selected players will be right-handed?

Just one more card…2. Hockey cards, chosen at random from a set of 20, are given away inside cereal boxes. Stan needs one more card to complete his set so he buys five boxes of cereal. What is the probability that he will complete his set?

Let’s play ball!3. While pitching for the Toronto Blue Jays, 4 of every 7 pitches Juan Guzman threw in the first 5 innings were strikes. What is the probability that 3 of the next 4 pitches will be strikes?

I don’t know this stuff!4. A true-false test has 12 questions. Suppose you guess all 12. What is the probability of exactly seven correct answers?

I won!!!!5. As part of a promotion, a store is giving away scratch-off cards. Each card has a 40% chance of awarding a prize. Suppose you have 5 cards. Find the probability that exactly four of the five cards will reveal a prize.

I’ve been hacked.

6a. Privacy is a concern for many users of the Internet. One survey showed that 59% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 10 Internet users, 6 are concerned about the privacy of their e-mail?

At leastIn many cases, we will be interested in the probability of a rather than in the probability of an exact number of successes. ex: What is the probability that 6 of the 10 Internet users have concerns about privacy.

I’ve been hacked again. 6b. Privacy is a concern for many users of the Internet. One survey showed that 59% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 10 Internet users, at least 6 are concerned about the privacy of their e-mail?

Ketchup 7. A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants 10 seeds.

a) What is the probability that exactly 8 seeds will germinate?

b) What is the probability that at least 8 seeds will germinate?

8. Sociologists say that 90%of married women claim that their husband’s mother is a horrible person. Suppose that six married women are having coffee together one morning. What is the probability that:

a) All of them dislike their mother-in-law?

b) none of them dislike their mother-in-law?

c) at least four of them dislike their mother- in-law?

d) no more than three of them dislike their mother in-law?

9. It is reported that 77% of workers aged 16 and over drive to work alone. Choose 8 workers at random. Find the probability that

a) all drive to work aloneb) 4 or more drive to work alonec) Exactly 3 drive to work alone

5.4 Normal Distribution

The term bell curve is used to describe the mathematical concept called normal distribution. The center contains the greatest number of a value and therefore would be the highest point on the arc of the line. This point is referred to the _____________ but in simple terms it is the highest number of occurrences of an element.

The important things to note about a normal distribution is the curve is ______________________in the center and decreases on either side.

This is significant in that the data has less of a tendency to produce unusually extreme values, called outliers, as compared to other distributions. 

A bell curve graph depends on two factors, __________________________________________.

There are two symbols each that can represent the mean and the standard deviation. mean –

standard deviation –

When we have a normal distribution with mean µ and standard deviation σ, we use the notation: _____________1. Write the notation for a normal distribution with a mean of 0 and standard deviation of 1.

68-95-99.7 Rule In terms of probability, once we determine that the data is normally distributed (bell

curved) and we calculate the mean and standard deviation, we are able to determine the probability that a single data point will fall within a given range of possibilities.

___________of the data values fall within _____ standard deviation of the mean in either direction

___________of the data values fall within _____ standard deviation of the mean in either direction

___________of the data values fall within _____ standard deviation of the mean in either direction

Examples:1. A Calculus exam is given to 500 students. The scores have a normal distribution with a mean of 78 and a standard deviation of 5. Give the ranges of exam scores centered on the mean that includes 68%, 95%, and 99.7% of the Calculus scores.

b) What percent of the students have scores are between 82 and 90?

c) How many students have scores between 82 and 90?

d) What percent of the students have scores between 72 and 90?

e) How many students’ scores were between a 72 and 90?

f) What percent of the students have scores above 60?

g) What percent of the students have scores above 70? And how many students scored about a 70.

2. Find the probability of scoring below a 1400 on the SAT if the scores are normal distributed with a mean of 1500 and a standard deviation of 200.

b) If the SAT if the scores are normal distributed with a mean of 1500 and a standard deviation of 200.Give a range of scores centered on the mean that includes 95% of the SAT scores.

Normal Distribution Practice: In the United States, the average height of an adult male is 5’9” (69 inches). Male height is distributed according to a normal distribution curve with a standard deviation of approximately 2.5 inches. Draw a normal distribution curve and shade accordingly.

1. What percent of heights falls below 66.5”?

2. What percent of the population falls below 69”?

3. What percent of the height lies within 2 standard deviations of the mean?

4. What percent of the heights lies within 1 standard deviation above the mean?

5. A male with a height of 71.5” is taller than what percent of the population?

6. Of 200 males, how many men can be expected to have a height between 69” and 71.5”?

7. Of 100 males, how many men can be expected to have a height between 71.5” and 74”?