Probability and Statistics – Mrs. Leahy

Unit 4: Elementary Probability Theory

Section 1 (Day 1): What is Probability?

Probability measures how ___________________ an event is to occur. Probability is measured between 0 and 1 and are frequently measured as fractions, decimals, and percents.

P(A) is read “P of A,” and denotes the probability of event A.

If:P(A) = 1 P(A) = 0

A is _________________ to occur. A ________________________ to occur.

Probability Assignments – Three Types:

1. INTUITION/PERSONAL/SUBJECTIVE Probability assignment based on intuition

incorporates past _______________, __________________, or ___________________ to estimate the likelihood of an event.

2. RELATIVE FREQUENCY/EXPERIMENTAL

Probability based on the formula: P(Event) = relative frequency = fn

where f is the frequency of the event in a sample of n observations.

3. EQUALLY LIKELY OUTCOMES/ CLASSICAL/THEORETICAL

Probability based on the formula:

P(event) = Number of outcomes favorable¿event ¿Total numberof outcomes

Example 1: Consider each of the following events, and determine how the probability is assigned.Subjective (Personal opinion)? Experimental (Relative Frequency)? Theoretical (Equally Likely Outcomes)?

a) Henry figures that if he guesses on a true-false question, the probability of getting it right is 0.50.

b) A sports announcer claims that Sheila has a 90% chance of breaking the record in the 100-yard dash.

c) The Right to Health Lobby claims that the probability of getting an erroneous medical laboratory report is 0.40, based on a random sample of 200 laboratory reports, of which 80 were erroneous.

Example 2: Assign a probability to the indicated event on the basis of the information provided. Indicate the technique you used: intuition, relative frequency, or the formula for equally likely outcomes.

a) A random sample of 500 students at Hudson College were surveyed, and it was determined that 375 wore glasses or contact lenses. Estimate the probability that a Hudson College student selected at random wears corrective lenses.

b) The Friends of the Library host a fund raising barbecue. George is on the cleanup committee. There are four members on this committee, and they draw lots to see who will clean the grills. What is the probability that George will be assigned the cleaning job?

c) Joanna photographs whales for Sea Life Adventure Films. On her next expedition, she is to film blue whales feeding. Based on her knowledge of the habits of blue whales, she is almost certain she will be successful. What specific number do you suppose she estimates for the probability of success?

Experimental Probability vs. Theoretical Probability

Based on ACTUAL data, things you count… Based on THEORETICAL data, things you think about…Also Called: Empirical Probability Also Called: Classical ProbabilityFormula: Relative Frequency Formula: Equally Likely Outcomes

Example: You flip a coin 50 times and get heads 28 times. You flip a coin.P(head) = P(head) =

The Law of Large Numbers *Figure 14.1 pg 325 text*In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or actual) probability value.

Example 3: Toss a coin repeatedly. The relative frequency gets closer and closer to P(head) = 0.50.

Important Terminology:

Statistical/Random Experiment: any random activity that results in a definite outcomeEvent: A collection of one or more outcomes of an statistical experimentSimple Event: one particular outcome of a statistical experimentSample Space: the set of all simple events

Using Sample Space

Example 4: You roll a die two times.

How large is the Sample Space?

a) P(sum of six) b) P(sum 7 or 11) c) P(sum of eleven)

d) P(doubles or sum of 4) e) P(sum of seven) f) P (sum greater than 10)

Example 5: Assign a probability to each following event based on the information provided.

a) A random sample of 400 students at a college were surveyed and 300 of them liked the food in the cafeteria. Estimate the probability that a student randomly selected at that college likes the cafeteria food.

b) Toss a fair die. What is the probability that you get a number less than or equal to 4?

c) John is to take a driver’s license test. Based on his knowledge and skills, John is almost sure that he will pass the test. What would he say about his probability of passing?

d) Randomly toss two fair coins. What is the sample space of this experiment? What is the probability that you get two heads? What is the probability of at least one head?

Probability Rules:

Example 6:The probability that a student who has not received a flu shot will get the flu is 0.45.

What is the complement of “will get the flu” ?

What is the probability of the complement?

P(A) + P(Ac) = 1

Example 7:A veterinarian tells you that if you breed two cream-colored guinea pigs, the probability that an off-spring will be pure white is 0.25. What is the probability that an offspring will not be pure white?

Example 8: Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates?

a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common.

b) Do the probabilities add up to 1? Why should they? What is the sample space in this problem?

c) Find P(at least 2) d) Find P(no more than 1)

Probability Related to Statistics

Statistics and Probability are related, but _____________________ mathematical fields.

Probability makes statements about what will occur when samples are drawn from ______________ populations.

Statistics describes to collect data how inferences are made about _________________ populations.

Example: ProbabilityWe KNOW the exact makeup of the ENTIRE population.Given 3 green marbles, 5 red marbles, and 4 white marbles in a bag. You draw 6 marbles from the bag. What is the probability that none of the marbles drawn is red?

Example: Statistics

We have only SAMPLES from an otherwise UNKNOWN population.You randomly draw 6 marbles from a bag of marbles without looking inside. Based on the sample, make a conjecture about the colors and numbers of marbles in the entire population of all the marbles in the bag.

Example 9:Can you roll your tongue?

Number of students in class: ________

Number of students that can roll their tongue: ________

What is the probability that a student chosen at random:

P(can roll tongue) =

P(cannot roll tongue) =

According to http://udel.edu/~mcdonald/mythtongueroll.html , National statistics indicate that about 65% - 81% of people can roll their tongue.

Unit 4: Section 2 Probability Rules – Compound Events

A few basic ideas:

Example 1: You roll a standard die one time.

a) P(even and prime)

b) P (even or prime)

c) P (not even)

d) P (even and less than 5)

e) P (even or less than 5)

f) P(less than 2 or more than 5)

g) P(less than 2 and more than 5)

Mutually Exclusive Events

Two events are mutually exclusive or _________________ if there are no elements in common to both sets.In other words:

P(A and B occurring at the same time) =

Example: P(King or Queen) P(King or Heart)

Don’t stress out about these formulas! Just think about what they are saying:When you are counting ‘or’, be really careful about counting overlap!

Example 6: Suppose you are drawing one card from a well-shuffled deck of 52 cards. *See Cards and Dice Handout*

a) Are the events “Ace” and “Clubs” mutually exclusive? What about “Hearts” and “Spades”?

b) P(Ace and Clubs) c) P (Ace or Clubs)

d) P (not Ace) e) P(hearts or spades)

f) P(King or Spades) g) P(Ace or Black Card)

Example 8: Suppose you are throwing two fair dice.

a) What is the probability that you will roll a sum bigger than 10?

b) What is the probability that you will roll doubles or a sum bigger than 10?

Example 9: Suppose you are throwing two fair dice.

a) P(sum of 3) b) What if we throw the white die BEFORE the black die? If we KNOW the white die is a 1, then we have a smaller sample space.

This is called CONDITIONAL PROBABILITY.

Further Examples (CONDITIONAL PROBABILITY)Tip: If you have enough data, just try to think of this as looking at probability in a REDUCED sample space.

A _____________________ is a survey that is formed by questions to which the responses can be recorded in a

table called a ____________________________ table.

Example 9: All 700 students at a college were surveyed about their residency status (live on campus, live off campus but in town, or commute). The students were grouped by gender. The results are showing in the following contingency table.

Suppose a student is selected at random from these 7000 students. Consider these events:M = male student F = female student C = on campus O = off campus T = commute

a) Find P(C) b) Find P(M)

c) Find P( ) d) Find P ( )

Suppose a student is selected at random from these 7000 students. Consider these events:

You may want a highlighter for these problems…

M = male student F = female studentC = on campusO = off campus T = commute

1) Find P(O) 2) Find P(F)

3) Find P(O and F) 4) Find P(O or F). Are the events mutually exclusive?

5) Find P( ) 6) Find P ( )

7) Find P( ) 8) Find P( )

Unit 4: Section 3:MULTIPLE EVENTS; Independent and Dependent Events

What if we are rolling a die THREE times or drawing TWO cards, one right after the other…

Two events are independent if the occurrence or

nonoccurrence of one does not ______________ the

probability that the other will occur.

Example: You roll a standard die and you flip a coin.

1. P(even number and heads) 2. P(4 and T)

Example: Select a card from a deck of cards. Replace the card. Select a second card.

3. P(Queen and Ace) 4. P (2 Queens)

WAIT WAIT WAIT… Mrs. Leahy, YOU SAID on DAY 2 that P(Queen and Ace) = 0 because a card couldn’t be a queen AND an ace. What’s up with that?!?!

PROBABILITY SITUATION 1: Compound EventONE THING is happening or multiple things are happening SIMULTANEOUSLY.

You draw 1 card. You roll 2 dice at the same time.event: (Queen and Heart) = 1 outcome, the Queen of Hearts event: (sum of 3) = 2 outcomes, (1,2) and (2,1)P(Queen and Heart) = 1/52 P(sum of 3) = 2/36

PROBABILTY SITUATION 2: Multiple EventSEPARATE THINGS are happening or one thing is happening REPITITIOUSLY.

You draw 2 cards, one after another with replacement You flip a coin three times.event: (Queen and Heart) means (Queen 1st, Heart 2nd) event: (3 Heads) means (Heads, Heads, Heads)P(Queen and Heart) = P(Queen, Heart) = 4/52 x 13/52 P(3 Heads) = P(Heads, Heads, Heads) = ½ x ½ x ½

Suppose you are drawing two cards without replacement from a well-shuffled deck of 52 cards. What is the probability of getting two hearts?

OH, OK… Continue…Two events are dependent if the probability of one event depends upon the occurrence of the other event.

Example:Select a card from a deck of cards. Without replacing it, select a second card.P(2 Queens) = ?

In order to calculate the probability of a dependent event, you need to understand Conditional Probability”

Conditional Probability

Conditional probability is the probability that a dependent event will occur given that another event has occurred.

Notation: = P (A will occur given that B has occurred)

EXAMPLES:Example 1: Example 2:Suppose you are going to throw two fair dice. What is the probability of getting a 3 on each die?

Example 3:John and Lisa are going to take the driver’s license test. Based on their skills, the probability that John will pass the test is 0.7 and the probability that Lisa will pass is 0.8. Assuming that the test result of one of them has no effect on the result of the other…

a) Are the events John will pass and Lisa will pass dependent or independent?

b) What is the probability that both will pass?

Don’t stress out about this formula! Just keep in mind that when you are dealing with MULTIPLE events, sometimes your sample space changes before you make your second choice.

Example 4: Suppose you are drawing two cards without replacement from a well shuffled deck of 52 cards.

a) What is the probability of getting a king on the first draw?

b) Suppose the first draw is indeed a king. What is the probability of getting a king again on the second draw?

c) Are the answers to (a) and (b) different?

d) What is the probability that both draws yield kings?

Example 5: You pack a cooler for the beach and stock it with soda cans. You include a 12 pack of cola, a 12 pack of diet cola, and a 6 pack of red bull energy drink.

You randomly grab 3 cans in a row and hand them to friends. What is the probability that you:

a) select 3 red bull energy drinks

b) select a cola, a diet cola, and a red bull (in that order)

Example 6: Your teacher is trying to demonstrate a probability concept and is running out of ideas. She creates three bags and fills them with 10 scrabble tiles.

First Bag:

Second Bag:

Third Bag:

Example 7: You consistently hit 80% of your free throw shots when you practice. What is the probability of hitting the next three shots?

Unit 4, Section 4: Trees and Counting Techniques

a) You draw tiles out of the first bag, one after the other, without replacement.

P(3 vowels) =

b) You draw one tile out of each bag.

P(3 vowels) =

c) You draw two tiles out of the second bag, one after the other, without replacement

P(not drawing an E at all) =

Multiplication Rule of Counting:

Example 1: You toss a coin three times. How many possible outcomes are there?

Tree Diagrams: A tree diagram is a visual way to represent the possible outcomes of multiple events.

Example 3: a) Suppose you are tossing two fair coins. Use a tree diagram to show all possible outcomes.

---------------------------- Suppose you take a survey and find that 25% of respondents are seniors. Of the seniors, 48% claim to be taking dual credit courses. What is the probability that a survey selected at random will be from a senior who is taking dual credit courses?

Example 4: Suppose you are drawing two cards with replacement from a well shuffled deck of 52 cards. You are to note the color of the first card (red or black), put it back, and then note the color of the second card.

Example 2: There are three reference books, five cookbooks, and four novels on the shelf. Jeff is going to pick one book of each kind. How many different sets are possible?

b) Now use the multiplication rule to compute the total number of outcomes in this experiment.

a) What are the outcomes of this experiment?

What is the probability of each outcome?

b) What if there is no replacement? NOW what is the probability of each outcome?

Example 5: Place 5 marbles in an urn: 3 red and two green. You draw one marble and note the color. Without replacing it, you draw a second ball and note the color.

a) Draw a tree diagram to represent this situation. How many outcomes are there?

b) What is the probability of each outcome?

Example 6: You have 4 cards labeled ‘A’ “B” “C” and “D”.

You draw a card, note the letter, and put it back. Then you draw a second card.

a) Draw a tree diagram. How many outcomes are there?

b) What is the probability of each outcome?

Factorials and Permutations

For a number n>0, its factorial n! =

By special definition: 0! =

Example 5: a) Evaluate 5! b) In how many ways can five objects be arranged in order?

Permutation:

Example 6:

Example 7: For a group of nine people, how many ways can six of them be seated into six chairs?

Combinations

Example 8:

Example 9: There are eight different books on the shelf. How many different groups of three books can be selected from the shelf?

For a group of seven people, how many ways can four of them be seated in four chairs?a). 35 b). 3 c). 28 d). 840

Among eleven people, how many ways can eight of them be chosen to be seated?

a). 6,652,800 b). 165c). 3 d). 88

Visually: Counting principle, Permutation, Combinations

Counting Principle Permutation Combination

Examples

Example 10: From a student organization of 20 members, three officers – president, vice president, and treasurer – must be elected. How many different slates of officers are possible?

a) Are the slates permutations or combinations?

b) How many different slates are possible?

Example 11: From a student organization of 20 members, three members will be selected to attend a convention. How many different groups of 3 are there?

a) Are the groups permutations or combinations?

b) How many different groups are possible?

Example 12: You are buying a new car and have the choice of three different wheel types, two different security packages, and five different colors. How many different combinations of one wheel type, one security package, and one color are possible?

PUTTING IT ALL TOGETHER: UNIT 4 REVIEW

The Formulas:

Examples:

1. The cup on my desk contains 3 red pens, 4 blue pens, 2 black pens, 2 green pens, and 1 orange pen. If I randomly select one pen to grade with, find the following probabilities.

a) P(green) b) P(not red) c) P(purple)

2. You roll a fair die. What is the probability of rolling a number of at least a 4 on a single throw?

3. You roll two dice: one is black and one is whiteWhat is the probability of rolling an even number on the black die and a 5 on the white die?

4. You write the 26 letters of the English Alphabet on individual cards and shuffle them. Vowels: AEIOU Consonants: BCDFGHJKLMNPQRSTVWXYZ

a) You draw two cards from the deck, but replace the first card before drawing the second card. What is the probability that the first card is a vowel and the second card is “W, X, Y, or Z” ?

b) You draw two cards from the deck, but do not replace the first card before drawing the second card. What is the probability that the first card is a vowel and the second card is “W, X, Y, or Z” ?

5. In your stats class, you learn that 30% of students got an A on the last test. Of the students who got an A, 85% of them had no missing homework assignments. What is the probability that a student selected at random from the class got and A on the last test and had no missing homework assignments?

6. Your teacher gives you a choice of four reading passages and 2 short essay questions. How many different ways are there to choose one of the reading passages and one of the essay questions? Draw a tree diagram to show the possible outcomes.

7. Adults and Children were surveyed and asked to state if they preferred chocolate or vanilla ice cream. The results are shown in the table.

If a survey participant is selected at random, what are the probabilities of the following:

a) P(Vanilla)

b) P(Chocolate and Child)

c) P(Child)

d) P(Adult|Chocolate)

e) P(Vanilla|Adult)

f) Are “Adult” and “Chocolate” independent events? Show the test for independence to support your statement.

8. A nail salon has 14 new colors of polish. How many ways can 5 different polishes be chosen to put on a display for the front window?

9. The band boosters sell 75 raffle tickets. First place wins $100, Second place $25, and 3 rd place wins a coupon for Taco Bell. How many ways can a 1st, 2nd, and 3rd place winner be chosen?

10. Mrs. Leahy has a collection of 10 corny jokes, 8 random anecdotes, and 5 bizarre movie references. How many ways can she choose 1 joke, 1 anecdote, and 1 reference to tell her long suffering math students?