Math 3307

Module 2: Probability and Making Predictions

Probability of an EventCounting Methods

Addition PrincipleMultiplication PrincipleFundamental Counting PrinciplePermutations and Combinations

Bernoulli trials and Binomial DistributionGeometric ProbabilityNormal Distribution

Measurement ScalesRandom SamplingThe Law of Large Numbers and The Central Limit TheoremHypothesis TestingRegression Analysis

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Probability of an Event

The probability of an event is the chance that an event will occur. A certain even has a probability of 1; something that cannot happen has a probability of 0; all other types of events have some proper fraction (often written as a decimal or percent) signifying the probability of happening.

P(event) =

Example: the 3 kid family. List the outcomes:

BBB GGBBBG GBGBGB BGGGBB GGG

How do I know I’ve listed all outcomes (hint: counting methods!)What method might illustrate this effective (hint: tree diagrams!)

If our event is have a daughter…then there are 7 out of 8 families that have a daughter.If our event is have exactly one daughter…then there are 3 of 8 families that work.If our event is having an oldest daughter…then there are 4 of 8 families that are fine.

CARE is needed in defining our event. “a” and “exactly one” are NOT the same!Watch for words like “at least one”…1 or 2 or 3.

Precision in language is crucial in probability.

Probability Problem 1

Ms. Shapiro owns 3 computer stocks (IBM, Honeywell, and Apple) and 2 oil company stocks (Exxon and Shell). She needs some cash and instructs her broker to sell all her shares of one stock and she doesn’t care which one. What is the probability the broker will sell an oil stock?

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Probability Problem 2

Suppose she has asked the broker to sell 2 stocks of his own choosing? What is the sample space? What is the probability that he sells one oil stock and one computer stock?

Probability Problem 3

Vincent and Joey are playing a game where they each display their right hand with a number of fingers showing simultaneously. (1, 2, 3, or 4 fingers).

What are the outcomes in the sample space?

What is the probability that they each display the SAME number of fingers?

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Probability Problem 3

Given the following probabilities for a number on a LOADED die:

Event Probability

1 1/32 1/43 1/124 1/125 1/66 1/12

Why is it necessary that all the Probabilities add up to one? Do these?

Find the probabilities of the following events:

A. the number observed is a multiple of 3

B. the number is even

C. the number is an even multiple of 3

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Probability Problem 5

A subcommittee of two is to be formed from a group consisting of 5 people: Juan, Dick, Mary, Paul, and Vinette. Write the following events as sets, listing the outcomes that they contain.

A. the sample space

B. both committee members are males

C. At least one member is male

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Probability Problem 6

Use a tree diagram to analyze the following experiment:

You toss a penny, a quarter, and a tetrahedral die (ie 4 sides!). You record the observable face in that order.

A. what is the sample space?

B. what is the probability that a 3 is on the face of the die?

C. what is the probability that 2 heads appear on the coins?

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Probability Problem 7

A letter of the English alphabet is chosen a random. Find the following probabilities:

A. the letter is a vowel

B. it follows the letter P in the alphabet

C. it follows the letter p and is a vowel

D. it follows the letter p or is a vowel

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E. it is a consonant? What is the quickest way to calculate this?

Probabilitiy Problem 8

Four hundred people attend an event and each is issued a number from 1 – 400 as they enter the room. A number is selected randomly. What is the probability that it is

A. 123

B. has the same digit repeated 3 times

C. ends in a 9

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Probability Problem 9

A person decides to take a vacation and cannot decide between Las Vegas, the Bahamas, and Hawaii. She is twice as likely to go to Hawaii as to Las Vegas and 3 times as likely to go to Las Vegas as to the Bahamas. Find the probability that she vacations in Las Vegas.

Probability Problem 10

Suppose the following probabilities are assigned to these events

P({e1}) = 3P({e2}) = P({e3}) and p({e4}) = 4P({e2})

A. find the probabilities assigned to each individual event

B. Find P(A) where A = {e1, e3, e4}

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Independent events

Two events are independent if the occurrence of one does not affect the occurrence of the other. Having children is independent. Tossing a fair die is independent. Rainfall over a weekend is NOT independent.

If 2 events are independent then the probability of both occurring is the product of the individual events.

Example: Getting 3 heads in a row when tossing a fair coin. P(one head in one toss) = 50%

P(head1 and then head 2 and then head 3) = (50%)^3 = 1/8

See “having an all girl” family above. To a statistician tossing coins and having kids are indistinguishable processes.

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PIE Problem 1

If 2 coins are tossed simultaneously, what is the probability that they’re both the same side after the toss?

PIE Problem 2

Using a tree diagram, find the probability of all three balls being black given that the urn contains 5 white balls and 7 black balls and you’re drawing them one at a time.

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PIE Problem 3

A bag contains 3 red, 4 green, and 5 blue balls. Three balls are drawn in succession and after each ball is drawn and the color recorded it is returned to the bag. What is the probability of drawing first a red, then a white, and then a blue ball?

PIE Problem 4

Two safes have 3 compartments each. A treasure map is hidden in one of the 6 compartments. You have 1 try to get to the right compartment. What is the probability that you get the map?

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PIE Problem 5

A certain number of playing cards – some spades, some hearts, some diamonds, and some clubs – are laid on a table face down. Suppose when picking a card the following are true:

The probability of picking a space is twice that of picking a heart.The probability of picking a heart is 3 times that of picking a diamond.The probability of picking a diamond is 4 times that of picking a club.

Find the probabilities of the following events:

A. the card is red

B. the card is a spade or a diamond

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PIE Problem 6

Past attendance records show that the probability that the chair of the board attends the annual sales meeting is 0.65, that the president of the company attends is 0.9, and that they both attend is 0.6. Would you say that they are acting independently? Show the basis of your reasoning.

PIE Problem 7

John Barry flies from DC to Portland via Chicago. The probability of landing safely in Chicago is 0.995 and, after switching to a second airline in Chicago, the probability of landing safely in Portland is 0.998. Determine the probability of the following events:

A. He lands safely on both legs of the journey.

B. He lands safely in Chicago but has a landing mishap (non-fatal) in Portland.

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PIE Problem 8

A number is picked at random from the digits {1, 2, 3, ….,9}. Then a fair coin is tossed and the observable side recorded. Then a fair die is tossed and the down side is recorded.

What is the probability of (even, heads, multiple of 3)?

PIE Problem 9

It is known from old medical records that the probability that a person has cancer is 0.02 and the probability of having heart disease is 0.05. Assume that the two ailments are independent events. What is the probability of having both? Neither? One or the other but NOT both?

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PIE Problem 10

A nuclear power plant has a fail-safe mechanism with 6 protective devices that function independently. The respective probabilities are 0.3, 0.2, 0.2, 0.2, 0.1, and 0.1. What is the probability that the reactor will fail?

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Conditional Probability

When the occurrence of one event AFFECTS the occurrence of the next, you have dependent events. This is called conditional probability. Essentially, the universe is CHANGED between the first and the second event.

Drawing cards with replacement is independent activity.Drawing cards WITHOUT replacement is dependent activity because you had 52 cards on the first draw and only 51 cards for the second draw.

Given that A and B are two dependent events is

The probability of (A and B) = P(A) times P(B given that A happened)

In symbols:

You may be asked to solve for any one probability having been given information on the other two probabilities.

CP Problem 1

A continuation of

Vincent and Joey are playing a game where they each display their right hand with a number of fingers showing simultaneously. (1, 2, 3, or 4 fingers).

What is the probability that they both display 2 fingers given that they display the same number of fingers?

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CP Problem 2

The probability that the stock market goes up on a Monday is 0.6 and the probability that it goes up on a Tuesday given that it went up on Monday is 0.3. Find the probability that the market goes up both days.

CP Problem 3

A cosmetics sales person makes house calls to different neighborhoods. Past experience has shown that the probability that a female resident is home when she calls in the evening is 0.7. Given that the lady of the house is home, the probability of a sale is 0.3. Find the probability that she’s home and she buys a product.

CP Problem 4

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Counting Methods

Addition Principle

Since we need to make a ratio of the number of our events and the number of possible outcomes, it becomes important to COUNT these events.

Given A and B are events, the number of outcomes in A and in B or in both is the number of outcomes in A plus the number of outcomes in B minus the number of shared outcomes that got counted once in A and again in B.

In symbols:

If you need a review of union and intersection NOW is the time to speak up!

The usual Venn diagrams might be helpful here:

The circle on the left is A and the one on the right is B; the box containing the circles is the Sample Space or the Universe.

Find A∩B, AUB, only A, neither A nor B…

Translate A∩B into words…what is the key word?

Translate AUB into words…what is the key word?

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Example:

In how many ways can you select a face card OR a black card from a standard deck of cards?

F = select a face card (King, Queen, Jack times 4 suits) nF = 12B = select a black card nB = 13

F intersect B (Face card that is also a black card) nF∩B = 6

12 + 13 – 6 = 19

P(selecting a face card or a black card on one draw) = 19/52

CM – A Problem 1

On a TV quiz show a contestant is asked to pick an integer at random from the first 100 natural numbers. Unbeknowst to the contestant, the rule for winning is to pick a number that is divisible by 9 or 12. If a winning number is picked, the contestant will win a trip to the Bahamas.

What is the probability that the contestant will win?

CM – A Problem 2

In a certain area, TV Channels 4 and 7 are affiliated with the same national network The probability that channel 4 carries a certain sports program is 0.5, that channel 7 carries it is 0.7 and the probability that they both carry it is 0.3. What is the probability that Mike will be able to watch the program on either channel? (ie Channel 4 OR Channel 7).

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CM – A Problem 3

The town of Alvin has 2 ambulance services: the city service and a private company. In an emergency, the probability that the city service responds is 0.6, the probability that the private company responds is 0.8, and the probability that either responds is 0.9. Find the probability that both services will respond to an emergency.

CM – A Problem 4

A box contains 3 red, 4 green, and 5 white balls. One ball is picked at random. What is the probability that it is red or white?

What’s different about this situation from the preceding ones? How will you use the formula:

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CM – A Problem 5

Two employees, Tom from Plant Operations and Becky from Campus Security, are supposed to check that the University Computing Center doors are locked after 6pm. On any given day, the probability that Tom will check the doors is 0.96 and the probability that Becky will check is 0.98. The probability that they both check is 0.95. What is the probability that neither of them will check the doors?

CM – A Problem 6

The probability that a radioactive substance emits at least one particle during any given hour is 0.008. What is the probability that it does not emit any particle at all during any given hour?

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CM – A Problem 7

Explain why it is not possible to have two events, A and B, with either of the following probabilities.

A. P(A) = 0.6 and P(A and B) = 0.8

B. P(A) = 0.7 and P(A or B) = 0.6

CM – A Problem 8

If A and B are events with 0.6 and 0.7 as their respective probabilities. Further P(A or B) = 0.9. Find P(not (A and B)).

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CM – A Problem 9

Past records indicate that the probability that a new customer will open a checking account is 0.7 and the probability that this new customer will open a savings account is 0.4 while the probability of opening both is 0.25.

What is the probability that a new customer will open a savings account or a checking account?

What is the probability that the customer will open neither and opt for a different service entirely?

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Multiplication Principle of counting for Dependent Events

Recall:

Given that A and B are two dependent events is

The probability of (A and B) = P(A) times P(B given that A happened)

In symbols:

n(B│A) is the number of ways B can occur given that A has happened.

Example

In how many ways can 2 Aces be drawn from a standard deck WITHOUT replacement?

Now in one draw, you can pull an ace or not. Since we are NOT interested in the “not” scenario, let’s continue with the “ace” scenario.

Now pulling an ace: there are 4 ways to do this. And the probability of doing this is 4/52 which simplifies to 1/13.

Since we do NOT replace the ace, we now have 51 cards with 3 aces. There are 3 ways to pull an ace on the second draw (remember, we’re not caring about the “not” situation). So there is a 3/51 probability of pulling a SECOND ace once the first one is drawn. This is the conditional probability term in the equation above (P(B│A)).

Thus P(ace on first AND ace on second without replacement) = 1/13(3/51)

Multiplication Principle for counting INDEPENDENT events.

Given A and B are 2 independent events with A being the set of outcomes for the first event and B being the set of outcomes for the second event that happens. If they are truly independent then the probability that they both occur is the product of their individual probabilities and the number of outcomes in their intersection is the product of the individual outcomes.

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Example:

Suppose you are making student id tags to be assigned to students randomly. The id tags will be a two digit number followed by 2 letters of the alphabet…suppose further that no repetitions are allowed. This is a 4 stage process:

There are 26x25x9x8 ways to make the ids with no repetitions of letters and numbers.(46,800)

Fundamental Counting Principle:

This works for combining several tasks or steps in a process. Suppose there are m possible outcomes for Stage 1, n possible outcomes for Stage 2, and z possible outcomes for Stage 3. Then there are m x n x z outcomes from doing the 3 tasks in succession.

Example:

Rosemary has 5 blouses, 3 skirts, and 4 scarves. She can make 5(3)(4) different outfits from her wardrobe pieces.

FCP Problem 1 There are 4 trains from Zeobia to Alpheraz in the morning and 6 trains making the return trip in the evening. How many different round trips are possible from Z to A and back?

FCP Problem 2

From the digits 2, 5, 7, and 9 how many different three-digit numbers are possible with repetition? Without?

FCP Problem 3

In how many ways can 3 prizes be won by 11 competitors if no person may win more than 1 prize? If there is no restriction on the number of prizes an individual may win?

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Permutations and Combinations

Permutations are the number of possible arrangements when order matters. A permutation of n objects taken r at a time is computed using

So your answer is a number of outcomes. Note that you still have to calculate the other part of the probability ratio once you’ve done ONE permutation!

A permutation of 11 objects taken 4 at a time is calculated:

which is 7,920.

Combinations are the number of possible arrangements when order does NOT matter.A combination of n objects taken r at a time is computed using

Again, your answer is a number of outcomes that you will use in a probability ratio.

A combination of 11 objects taken 4 at a time is calculated

which is 330.

Note that there are far fewer combinations than permutations? Why is this always true?

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P&C Problem 1

How many 8-letter permutations can be formed from the letters in the word “hesitate”. In how many of these do the e’s occur side by side?

P&C Problem 2

Find the number of numbers greater than 4000 which can be written using the digits 0, 3, 5, 6, 8.

P&C Problem 3

Find the number of different lines determined by 11 points, no three of which lie on the same line.

P&C Problem 4

From a group of 12 students, in how many ways can a 6 person team be formed if a specific person MUST be included?

P&C Problem 5

Find the number of ways of arranging the letters in the word “object”. Find the number of ways of arranging the letters if the first letter must be “j”.

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P&C Problem 5

Would you use a permutation or a combination to solve the following problems:

A. the number of ways a 5 card hand can be dealt from a 52 card deck

B. making 3 digit numbers from the digits {1, 2, 3, 4, 5}

C.

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Bernoulli trials and probability

A Bernoulli trial is an experiment that has only 2 outcomes (yes/no, boy/girl, heads/tails, broken/working…). We establish a probability for ONE outcome and subtract it from 1 to get the probability of the other outcome.

If the probability of a washing machine straight from the factory being broken is 1/100, then the probability that it is fine is 99/100. P and (1 – p) are the outcomes.

In order to calculate statistics with Bernoulli trials the following situations must be present:

You run the experiment a fixed number of times (generally more than 10). This is called the n of the experiment (number of trials).

The n trials are all independent. The probability of a success is the SAME for each trial.

The classical experiment is tossing a coin: H/T, p is 50% for a head and 1-50% for a tail.The number of heads that we see is called X, the random variable in the trial. The distribution of X is called the binomial distribution.

Here’s why the Binomial Distribution shows up in this section:

If X has a binomial distribution with n trials and a probability p of success AND the outcomes for X observations are 0, 1, 2, …., n.

Then the probability that X takes on any particular value in the outcomes list, k, is calculated with

.

Suppose you have a loaded coin with the probability of tossing a head of .8. Then the probability of getting a tail is 1− .8 = .2. If you toss the coin 12 times, and let X = the number of heads observed in the 12 trials, what is the probability of getting 3 heads (3 = k in above).

Let’s look at this carefully and then do the calculation.

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BT Problem 1

The probability that a person who undergoes a kidney transplant successfully is 60%. Suppose we have 6 patients from the Medical Center, randomly selected, who each had a kidney transplant. What are the following probabilities:

None are successful

4 are successful

BT Problem 2

A test has 5 questions, and to pass the test, a student has to answer at least 4 questions correctly. Each question has 3 possible answers of which exactly 1 is the correct answer. If a student guesses on each question, what is the probability that the student will pass the test?

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BT – Problem 3

Suppose that 80% of the families in a rural county in Arkansas own a television set. If 10 families are interviewed at random, use the binomial probability formula to find the probability that

Seven families own a tv

At most 3 families own a tv

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BT – Problem 4

A drug manufacturing company is debating whether a vaccine is safe enough to be marketed. The company claims that the vaccine is 90 percent effective…which is to say that if the vaccine is used on a person, the chance that this person will develop immunity is 0.9. The FDA, however, believes that this claim is exaggerated and that the drug is only 40% effective. In order to come to a conclusion, the drug will be tested on 10 people. If 8 or more develop immunity, the company will get the go ahead.

Find the probability that:

The company claim will be granted incorrectly, which is to say that the FDA is right, but the sample doesn’t show it.

Hint: use p = .4 and see what it takes to get 8, 9, or 10 immune subjects…why does this work? (you should get 1.3%)

Or

The company claim will be denied incorrectly, which is to say that the company is right and the sample doesn’t show it.

Hint: use p = .9 and see what it takes to get 0 – 7 immune subjects…why does this work?(you should get 6.9%)

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Geometric Probability

Geometric Probability comes into play when working with geometric shapes and measures. It has a real place in a study of probability and statistics because we quickly transfer the notion of probability to AREA as we move deeper into the subject.

Here’s a simple example:

Given a piece of string 12 inches long, and a pair of scissors, if you’re cutting the string into 3” pieces or multiples of 3” pieces what’s the probability of getting a 9” length of string?

This can happen 2 ways:

First way: you cut the string right at the 3” mark from the left – or –Second way: you cut the string at the 3(3”) mark from the left.

Any other cuts will leave you with lengths that are NOT 9”.The cuts are at 3”, 6”, 9”

Thus 2/3 of the cuts will give you a 9” string. This is about 66% (or 67% if you want to round up). Note that we are assuming that you have NO PREFERENCE for cutting a 9” string and that you’re cutting choices are equally weighted.

Here’s another example.

There’s a big rectangular field (20’x 30’) that is being shared by some model plane enthusiasts and you and your true love. The enthusiasts are landing their planes where ever and when ever. You spread your blanket for a romantic picnic; your blanket is (4’x3’). What’s the probability of a plane landing on the blanket?

Well, the ratio of the area of your blanket (12’) and the field (600’) will tell you that. It’s 2%.

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The Normal Distribution

AKA The Bell Curve.

It is symmetric, mound-shaped and used quite frequently.

The area under the Bell Curve is 1 (as in 100%). To find the probability of a particular value being observed, you use the z-score of that observation and a table showing areas at or to the left of that measurement.

Example:

The Starbucks on my corner has a customer in and out pattern that follows a bell curve from 6am to 10am – it peaks at 8am. The mean number of customers an hour is 50 with a standard deviation of 10. Determine the probability that the number of customers tomorrow morning will be 40 or fewer.

First we calculate the z score for 40:

AHA! Now how much area is at or below a z-score of −1, one standard deviation below the mean? Well, 68.3 % of the data is between −1 and 1 standard deviation from the mean…let’s work it out!

P(X< 40) =

Distributions:

D Problem 1

The life span of a whale, measured in 20- year time spans has a probability density function of

0 – 20 years .3720 – 40 years .2240 – 60 years .1560 – 80 years .1480 – 100 years .12

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Sketch this curve. Find the probability that a randomly selected whale is

more than 60 years old, less than 20 years old, between 40 and 80 years old.

D Problem 2

A symmetric mound-shaped distribution is centered at X = 0. The probability that X is between −2 and 2 is 0.4 and the probability that X is greater than 3 is 0.18.Sketch this distribution.

Find the probabilities:

P(0 < X< 2)

P (X < −2)

P(2 < X < 3)

P( X < 2)

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D Problem 3

An isosceles triangle centered at 0 with a base length 1.0 describes the distribution of the error in ounces in filling a one pound bag of flour by a filling machine. Error is positive if there is overfill and negative if there is underfill. The apex of the triangle is at 2 on the y-axis.

First: Sketch the distribution:

Then, find the following probabilities:

The bag is

Overfilled by more than 0.2 ounces

Underfilled by more than 0.3 ounces

Underfilled by more than 0.2 ounces or overfilled by more than 0.3 ounces

Measurement Scales

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Random Sampling

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The Law of Large Numbers and The Central Limit Theorem

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Hypothesis Testing

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Regression Analysis

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