From Randomness to Probability

Randomness• We have learned to

understand randomness.

• The surprising fact is that in the long run even truly random phenomena settle down in a way that is consistent and predictable.

Dealing with Random Phenomena• Every day you drive through

the intersection at College and Main

• It seems like the light is never green when you get there.

• But this can’t really be true.

• In fact, if you try really hard, you can recall just sailing through the green light once in a while.

Random Phenomena• What is random here?

• The light itself is governed by a timer

• The light may be red at precisely the same times each day.

• It is the pattern of your driving that is random.

• At the precision level of 30 seconds or so that light spends being red or green, the time you arrive at the light is random.

• Even if you try to leave your house at exactly the same time every day, whether the light is red or green as you reach the intersection is a random phenomenon.

Random Phenomena• Is the color of the light

completely unpredictable?

• You expect some kind of regularity in the long-run experience.

• Some fraction of the time the light will be green as you get to the intersection.

• How can you figure out what that fraction is?

Random Phenomena• You might record what happens at the intersection

each day and graph the accumulated percentage of green lights like this:

Random Phenomena• The first day you

recorded the light, it was green. Then on the next five days: red, then green again, then green, red, and red.

• If you plot the percentage of green lights against days, the graph would start at 100%

• The next day it was red so the accumulated percentage of green dropped to 50% (1 out of 2)

• The third day it was green again (2 out of 3 or 67%)

• Then green again (3 out of 4 or 75%

Random PhenomenaDay Light % Green

1 Green 100

2 Red 50

3 Green 66.7

4 Green 75

5 Red 60

6 Red 50

… … …

• As you collect new data value for each day each new outcome becomes a smaller and smaller fraction of the accumulated experience

• In the long run, the graph settles down.

• As it settles down you can see that, in fact, the light is green about 35% of the time.

Random Phenomena• Let us define some terms:

• Each occasion upon which we observe a random phenomenon is called a trial

• At each trial, we note the value of the random phenomenon, and call that the trial’s outcome.

Random Phenomena• For the traffic light, there

are really three possible outcomes: red, yellow, green

• Often we are interested in a combination of outcomes rather than in the individual ones

• When you see a light turn yellow, what do you do?

• Do you gun it and try to catch it before it turns red?

• Or do you slow down and wait to safely catch the next green?

• No matter what you choose you may want to group the yellow with one or the other.

Random Phenomena• Whether you treat the

yellow light more like a red or a green, you may want to combine outcomes.

• Such a combination is called an event

• We sometimes talk about the collection of all possible outcomes and call that event the sample space

Random Phenomena - Terms• Observing a random

phenomenon is called a trial

• At each trial we note the value of the random phenomenon and call that the trial’s outcome

• We combine outcomes to create an event

• The collection of all possible outcomes is called the sample space.

• We denote the sample space S

• For our example, S = {red, green, yellow}

The Law of Large Numbers• What is the probability

of a green light at the intersection?

• It looks like the relative frequency settles down to about 35% so saying that the probability is .35 seems a reasonable answer.

• But do all phenomena behave well enough for this to make sense?

• Perhaps the relative frequency of an event can bounce back and forth between two values forever, never settling on just one number…

The Law of Large Numbers• Fortunately, a principle

called the Law of Large Numbers (LLN) gives us the guarantee we need.

• It simplifies things if we assume that the events are independent, the outcome of one trial doesn’t affect the outcomes of the others.

• The LLN says that as the number of independent trials increases the long run relative frequency of repeated events gets closer and closer to a single value.

• The LLN wasn’t proven until the 18th century

• “For even the most stupid of men…is convinced that the more observations have been made, the less danger there is of wandering from one’s goal.”

- Jacob Bernoulli, 1713, discoverer of the LLN

The Law of Large Numbers• Because the LLN

guarantees that relative frequencies settle down in the long run, we can now officially give a name to the value that they approach.

• We call it the probability of an event.

• If the relative frequency of green lights at the intersection settles down to 35% in the long run, we say that the probability of encountering a green light is 0.35 and we write:

• P(green) = 0.35

Probability• The value that events

approach in the long run is called the probability

• Because the definition is based on repeatedly observing the same outcome, this definition of probability is often called empirical probability.

• For an event A,

In the long run.

Non-Existent Law of Averages!• The LLN says nothing

about short term behavior

• The gambler’s fallacy that if a certain outcome hasn’t happened in a long time that it is now “due” is completely wrong!

• The LLN speaks only about long-term behavior.

Modeling Probability• It is easy to find

probabilities for events that are made up of several equally likely outcomes.

• We just count all the outcomes that the event contains. The probability of the event is the number of outcomes in the event divided by the total number of possible outcomes.

• For example, the probability of drawing a face card (Jack, Queen, King) from a deck of cards is

𝑃 ( 𝒇𝒂𝒄𝒆 𝒄𝒂𝒓𝒅 )=¿ 𝑓𝑎𝑐𝑒𝑐𝑎𝑟𝑑𝑠¿𝑐𝑎𝑟𝑑𝑠 =

1252=

313

Modeling Probability• Finding the probability of

an event when the outcomes are equally likely is straightforward, but it is not necessarily easy.

• It gets hard when the number of outcomes in the event (and the sample space) gets big.

Modeling Probability• Think about flipping two

coins.

• The sample space is:S = {HH, HT, TH, TT}

• Each outcome is equally likely so what is the probability of getting exactly one head and one tail?

• Let’s call that event A

• There are two outcomes in the event A = {HT, TH}

• This is out of four equally likely outcomes in S

• So

Modeling Probability• Now flip 100 coins.

• What is the chance of getting exactly 67 heads?

• There are 1,267,650,600,228,229,401,496,703,205,376 different outcomes possible when flipping 100 coins.

• No way this is going to be easy!

The First Three Rules of Working With Probability• 1) Make a Picture

• 2) Make a Picture

• 3) Make a Picture

• The most common type of picture is called a Venn diagram. We will use these a lot. Even experienced statisticians make Venn diagrams to help them think about probabilities of compound or overlapping events.

Formal Probability• We need to be precise!

Using “50/50” in conversation can mean “I don’t know” or “whatever”

• We will use it to mean equally likely.

Formal Probability• 1) If the probability is 0,

the event can’t occur

• If the probability is 1, the event always occurs.

A probability is a number between 0 and 1For any event A,

Formal Probability• 2) If a random

phenomenon has only one possible outcome, it is not very interesting or random.

• We need to distribute the probabilities among all the outcomes a trial can have.

• Probability Assignment Rule:

The set of all possible outcomes of a trial must have a probability 1

P(S) = 1

Formal Probability• 3) Suppose the probability

you get to class on time is 0.8

• What the probability that you do not get to class on time? Yes, it is 0.2

• The set of outcomes that are not in the event A is called the complement of A and is denoted

• Complement Rule:

The probability of an event occurring is 1 minus the probability that it does not occur:

For Example• Our traffic light is green

35% of the time.

• If P(green) = 0.35, what is the probability that the light is not green when you get to the intersection?

• P(not green) = 1 – P(green)

• =1-0.35 = 0.65

• There is a 65% chance that I won’t have a green light.

Formal Probability• 4) Suppose the

probability that (A) randomly selected student is a sophomore is 0.20

• The probability that (B) he or she is a junior is 0.30

• What is the probability that the student is either a sophomore or a junior, written: ?

• The Addition Rule says that you can add probabilities of events that are disjoint.

• To see whether events are disjoint we take them apart into their component outcomes and check whether they have any outcomes in common.

Formal Probability• 4 Continued:

• Disjoint (or mutually exclusive) events have no outcomes in common. The Addition Rule states:

• For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events.

• Provided A and B are disjoint

For Example• P(green) = 0.35

• Suppose we find out that P(yellow) is about 0.04

• What is the probability that the light is red?

• Red is the only remaining alternative and the probabilities must add up to 1 so:

Formal Probability• Be careful! The Addition

Rule does not work for events that are not disjoint.

• If the probability of owning an MP3 player is 0.50 and the probability of owning a computer is 0.90 the probability of owning both is not 1.40!

• The Addition Rule does not work, you cannot add these probabilities, because the events are not disjoint. You can own both!

Formal Probability• 5) Suppose your job

requires you to fly from Atlanta to Houston every Monday morning. The airline’s website reports that this flight is on time 85% of the time.

• What is the chance that it will be on time for two weeks in a row?

• That is, what is the probability that it is on time this week and on time next week.

• For independent events the answer is very simple.

Formal Probability• For two independent

events A and B, the probability that both A and B occur is the product of the probabilities of the two events.

Provided that A and B are independent

• This rule can be extended to more than two independent events. What is the chance of your flight being on time for a month – four Mondays in a row?

• We multiply the probabilities of it happening each week:

.85 x .85 x .85 x .85 = 0.522

Formal Probability Example• We have determined

that the probability that we encounter a green light at the intersection is 0.35, a yellow light is 0.04, and a red light is 0.61

• Let us think about your morning commute in the week ahead.

Formal Probability Example• P(green) = 0.35

P(yellow)= 0.04P(red) = 0.61

• What is the probability that you find the light red on both Monday and Tuesday?

Answer: Because the color of the light you see on Monday is independent of the color you see on Tuesday we can use the Multiplication Rule:

There is about a 37% chance it will be red on both Monday and Tuesday.

Formal Probability Example• What is the probability

that you do not encounter a red light until Wednesday?

• For that to happen you would have to see Green or Yellow on Monday and Tuesday and then a red on Wednesday.

• Simplify by thinking of it as not red on Monday and Tuesday then red on Wednesday:

Formal Probability Example

• There is about a 9% chance that this week I will hit my first red light on a Wednesday morning.

Formal Probability Example• What is the probability that

you will have to stop at least once during the week?

• This means I will have to stop for the light either 1, 2, 3, 4, or 5 times next week.

• It is easier to think about the complement: never having to stop at a red light.

Formal Probability Example

• There is a 99% chance that I will have to hit at least one red light some time this week.

• Hint: the phrase “at least” is often a tip-off to think about the complement.

Just Checking• Opinion polling

organizations contact respondents by telephone.

• Random telephone numbers are generated and interviewers try to contact those households.

• In the 1990’s this method could reach about 69% of US households

• According to the Pew Research Center for the People and the Press, by 2003 the contact rate had risen to about 76%. We can reasonably assume each households response to be independent of the others. What is the probability that…

Just Checking• P(contact in ‘90) = 0.69

P(contact in ‘03) = 0.76

• A) The interviewer successfully contacts the next household on her list?

• Answer:0.76

• B) The interviewer successfully contacts both of the next two households on her list?

• Answer:

• The interviewer has about a 58% chance of contacting the next two on her list.

Just Checking• The interviewer’s first

successful contact is the third household on the list?

• For this one, it is the probability of not contacting the first two then contacting the third.

• P(not contacting) = 1 – P(contacting)

• = 1 – 0.76= 0.24

• There is about a 4.3% chance of contacting someone only at the third call.

Just Checking• What is the probability

that the interviewer makes at least one successful contact among the next five households on the list?

• There is that “at least” again – time to start thinking about the complement.

• This is the probability of the caller making a successful call either on the 1st, 2nd, 3rd, 4th, or 5th call.

• This could be thought of as the probability of not making 5 unsuccessful calls.

Just Checking• P(make one successful call out of 5) = 1 – P(5 unsuccessful calls)

• There is a 99.9% chance that the caller will make a successful call in the first five attempts.

Another Step-By-Step Example• In 2001 the maker of

M&Ms decided to add another color to the standard lineup of brown, yellow, red, orange, blue, and green. To decide which color to add they surveyed people in nearly every country of the world and asked them to vote among purple, pink, and teal.

• The global winner was purple!

• In the US 42% of voters said purple, 37% said teal, and only 19% said pink

• In Japan the percentages were 38% pink, 36% teal, and only 16% purple

M&Ms• Japan: P(pink) = 0.38

P(teal) = 0.36 P(purple) = 0.16

• What is the probability that a Japanese M&Ms survey respondent selected at random preferred either pink or teal?

• Answer:We must first make sure the answers are legitimate. To be legitimate all the probabilities add up to 1.

• Here they only add up to 0.9, so the remaining respondents must have expressed no preference or written in another color.

M&Ms• Japan: P(pink) = 0.38

P(teal) = 0.36 P(purple) = 0.16 P(no pref.) =

0.10

• Question: What is the probability that a Japanese M&M survey respondent selected at random preferred either pink or teal?

• Answer: The events “pink” and “teal” are invidiual outcomes (could not choose both colors) and they are disjoint, so we may apply the addition rule

• The probability a respondent picked either pink or teal is 0.74

M&Ms• Japan: P(pink) = 0.38

P(teal) = 0.36 P(purple) = 0.16 P(no pref.) =

0.10

• If we pick two respondents at random, what is the probability they both said purple?

• Answer: The word “both” suggest we want which calls for the multiplication rule.

• P(both pick purple) =

M&Ms• If we pick three

respondents at random, what is the probability that at least one preferred purple?

• Here we again see the term “at least” and this clues us in to the fact that we may need to use the complement of a set,

• P(at least one picked purple) = 1 – P(none picked purple)

• There is about a 40.73% chance that at least one of the respondents picked purple

Homework• Page 338, # 1, 5, 9, 11

Page 339, #19, 21, 23, 25, 27