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Chapter 11: Probability and Statistics

11.1 Permutations and Combinations

The Fundamental Counting Principle allow us to count large numbers of possibilities quickly.

You can extend the idea to any number of choices.

Example 1: A college offers 3 different English courses, 5 different math course, 2 different art courses, and 4 different history courses. In how many ways could a student choose 1 of eachtype of course?

Frequently with counting problems you will use combinations of letters and digits.

There are ____ different digits possible:

There are ____ letters in the alphabet.

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Permutations

A permutation is an _______________________________ of items in a particular order.

Example 2: How many ways can 5 Scrabble tiles be placed in a row?

Any time you have a number in the form N x (N-1) x (N-2) x . . . x 3 x 2 x 1 you can write it as N! (read “N Factorial”)

Example 3 a) How many ways can 10 different b) How many ways can you arrangetextbooks be arranged on a shelf? 8 different shirts on hangers in your closet?

Example 4: Calculate the following a) 5! b) 15!

Example 5:Ten students are in a race. How many ways can 10 runners finish 1st, 2nd, and 3rd? (no ties allowed)

Method 1: Fundamental Counting Principle Method 2: nPr Formula

Most scientific calculators have and nPr function to compute the number of permutations of n objects taken r at a time.

COUNTING THE NUMBER OF PERMUTATIONSSometimes we want to arrange items from a set, but we don’t want to use ALL the items

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TI 83Type in “n” valuePress “MATH” button.Arrow over to PRBChoose option #2Type in “r” valueHit enter

n P r = 10 P 3

Example 6: Calculate the following

a) 12P8 b) 20P5

Permutations with Repetition

Example 7: How many ways can you arrange the letters of the word “PARTY” ?

Example 8: How many ways can you arrange the letters of the word “HAPPY”?

Example 9: How many ways can you arrange the letters of the word “MISSISSIPPI” ?

Example 10: How many ways can you make a stack of 5 blocks, 3 of which are red and 2 of which are blue?

Combinations

Repetition?

If you are assigning position to things and some of the items are identical…

Example: N itemsx are the samey are the same

#arrangements =

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A combination is a _____________________ of items in which order doesn’t necessarily matter.

Example 11:“How many ways can you select three tiles from a set of five?”

Example 12: Calculate the following

a) 15C3

b) 12C9

For a TI83, the nCr command is located right under the nPr command in the Math -> PRB menu

Example 13: For each problem, determine whether to solve using a permutation or a combination. Then find the solution.

a) A chemistry teacher divides his class into eight groups. Each group submits one drawing of the molecular structure of water. He will select four of the drawings to display. In how many different ways can he select the drawings?

b) You will draw winners from a total of 25 tickets in a raffle. The first ticket wins $100. The second ticket wins $50. The third ticket wins $10. In how many different ways can you draw the three winning tickets?

11.2 Probability

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Probability is how _________________ an event is to occur. The probability of an event is measured somewhere between 0 and 1.

P(impossible) = ______ P(certain) = ______

Many times probabilities are expressed as fractions or as percentages.

Actually collecting data:

Example 1:Of the 60 vehicles in the teachers’ parking lot today, 15 are pickup trucks. What is the experimental probability that a vehicle in the lot is a pickup truck?What is the probability that a vehicle is NOT a pickup truck?

Example 2:A softball player got a hit in 20 of her last 50 times at bat. What is the experimental probability that she will get a hit in her next at bat?

A simulation is a model of an event. You can use a simulation to find the experimental probability of an event.

***RANDINT HANDOUT***

From your simulation, how likely is it to pass a 10 question, 4 choice multiple choice test if you randomly guess at the answers?

The sample space is the set of all possible outcomes of an experiment.

When each outcome has the same chance of occurring, the outcomes are called equally likely outcomes.

Rolling a number cube:

Sample Space:

Equally likely?

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Example 3: Given a standard number cube, what is the theoretical probability of each event?

a) P(even) b) P(7) c) P(even or multiple of 3)

Example 4: Given a standard deck of 52 playing cards, what is the theoretical probability of each event?

Note: 13 Cards: Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,24 Suits: Black Suits – Clubs, Spades Red Suits – Diamonds, Hearts

a) P(King) b) P(Face Card) c) P(heart or a 10)

Example 5: You open a bag of jawbreakers and find 15 red, 10 orange, 8 green, and 7 purple. What is the theoretical probability of each event?

a) P (red or orange) b) P(not green) c) P(red or not orange)

Combinatorics

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When you are simultaneously choosing several things at once, you may need to use combinations to determine the probability of an event.

Example 6: What is the theoretical probability of being dealt the following 5 card hands from a standard 52-card deck?

a) P(exactly 2 kings) b) P(exactly 3 kings)

c) P (all 2’s or 3’s) c) P(5 diamonds)

Example 7: A student has a personal library of 30 zombie movies and 10 action movies. If the student randomly grabs 6 movies to take on vacation, what are the following theoretical probabilities?

a) P(all zombie movies) b) P(all action movies)

c) P(exactly 2 zombie movies) d) P(exactly 4 zombie movies)

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11.3 Probability of Multiple Events

Consider the following: I have a stack of 10 cards each with a different number from 1 to 10.1,2,3,4,5,6,7,8,9,10

Simple Probability (1 item, 1 time)

You draw 1 card from the stack. What is the probability that the card is an even number?

Combinatorics (several items at the same time)

You choose 3 cards from the stack. What is the probability that all three cards are even?

You choose 3 cards from the stack. What is the probability of getting exactly 2 odd-numbered cards?

Multiple Events (1 item, several trials or different items)

You roll a standard number cube and flip a coin. What is the probability of getting a even number and heads?

You flip a coin 6 times in a row. What is the probability of getting all heads?

Dependent vs. Independent

dependent - one event affects the outcome of another

independent – events do not affect each other

Example 1: a) Roll a number cube. Spin a spinner.

b) Draw a card. Without replacing it, draw a second card.

c) Flip a coin. Flip the coin again.

d) Select a coin from a pile. Put it back. Select a coin again.

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Probability of Independent Events

If A and B are independent events, then P(A and B) = P(A) x P(B)

Example 2: Find the probability of each of the following.

a) You are rolling a standard number cube then flipping a coin.

P( even number and head)

b) You are flipping a coin three times.

P(3 heads)

c) There are 10 diet cokes and 5 regular cokes. There are 8 bags of Doritos and 12 bags of Gardettos.

P( diet drink, Doritos)

P( regular drink, gardettos)

Mutually Exclusive Events Two events that cannot happen at the same time.

P(A and B) = 0

Example 3: You roll a standard number cube.

P(2 and 3) P(even and prime)

Example 4: You draw a card from a deck of 52.

P(K and Q) P(K and heart)

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“OR”Cannot happen at same time. CAN happen at same time.Mutually Exclusive Not Mutually ExclusiveP(A or B) = P(A) + P(B) P(A or B) = P(A) + P(B) – P(A and B)

Example 5: Students can take one foreign language at a time.37% of students take Spanish. 15% of students take French.

P(Spanish or French)

Example 6:Students can take multiple language courses at the same time.37% take Spanish. 15% take French. 5% take Spanish and French.

P(Spanish or French)

Example 7: You are working with a standard deck of 52 playing cards.

a) P(heart or a diamond) b) P(heart or a King)

c) P(Ace or a King) d) P (Ace or Black Card)

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11.4 Conditional Probability

The Monty Hall Problem

Monty Hall was the host of a classic game show called “Let’s make a deal.”

Classic Problem: A contestant is shown 3 possible doors. Behind 1 door is a new car. Behind the other two doors are goats.

The contestant gets to choose one of the doors.

After the contestant chooses a door, Monty opens up one of the OTHER doors to reveal a goat. The contestant is then given the chance to keep the door they chose, or to switch doors. What should the contestant choose?

Conditional Probability Exists when two events are DEPENDENT…

= “The probability of event B, given event A”

Example 1: The table shows students by gender at two- and four-year colleges, and graduate schools, in 2005. You pick a student at random.

a) ?

b)

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Example 2: Americans recycle increasing amounts through municipal waste collection. The table shows the collection of data for 2007.

a) What is the probability that a sample of recycled waste is paper?

b) What is the probability that a sample of recycled waste is plastic?

c) What the probability that a sample of recycled waste is glass?

There is also a formula for Conditional Probability

Example 3: A utility company asked 50 of its customers whether they pay their bills online or by mail. What is the probability that a customer pays the bill online, given that the customer is male?

Example 4: Researchers asked shampoo users whether they apply shampoo directly to their head, or indirectly with their hand. What is the probability that a respondent applies shampoo directly to their head, given that the respondent is female?

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NOTE:

Example 5: A school system compiled the following information from a survey it sent to people who were juniors 10 years earlier.

85% of the students graduated from high school Of the students who graduated from high school, 90% are happy with their jobs. Of the students who did not graduate from high school, 60% are happy with their jobs.

a) What is the probability that a person from the junior class 10 years ago graduated from high school and is happy with his or her job?

b) What is the probability that a student from the junior class 10 years ago did not graduate and is happy with his or her present job?

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11.9 Normal Distributions

Many common statistics (human height, weight, blood pressure) tend to have a “normal distribution” to their mean (their average). You may have heard this called a “bell curve”

Sometimes an extraordinary factor may cause the data to be “skewed”

Example 1: The bar graph gives the weights of a population of brown bears. The curve shows how the weights are normally distributed about the mean, 11.5kg.

a) Approximately what percent of female brown bears weigh between 100 and 129 kg.

b) Approximately what percent of female brown bears weigh less than 120 kg?

c) The standard deviation in the weights of female brown bears is about 10 kg. Approximately what percent of female brown bears have weights that are 1.5 standard deviations of the mean?

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Example 2: For a population of male European eels, the mean body length is 21.1 inches and the standard deviation is 4.7 inches. Sketch a normal curve showing eel lengths at one, two, and three standard deviations from the mean.

Example 3: The scores on an Algebra 2 final are approximately normally distributed with a mean of 150 and a standard deviation of 15.

a) What percentage of the students who took the test scored about 180?

b) If 250 students took the final, approximately how many scored above 135?

c) If 13.6% of the students received a B on the final, how can you describe their scores?