Slides developed by Mine Çetinkaya-Rundel of OpenIntro

The slides may be copied, edited, and/or shared via the CC BY-SA license

​

To make a copy of these slides, go to File > Download as > [option],�as shown below. Or if you are logged into a Google account, you can choose Make a copy... to create your own version in Google Drive.

Introduction

to Probability

​

Slides developed by Mine Çetinkaya-Rundel of OpenIntro

The slides may be copied, edited, and/or shared via the CC BY-SA license

Some images may be included under fair use guidelines (educational purposes)

Random processes

• A random process is a situation in which we know what outcomes could happen, but we don't know which particular outcome will happen.
• Examples: coin tosses, die rolls, iTunes shuffle, whether the stock market goes up or down tomorrow, etc.
• It can be helpful to model a process as random even if it is not truly random.

http://www.cnet.com.au/itunes-just-how-random-is-random-339274094.htm

Probability

There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow.

• P(A) = Probability of event A
• 0 ≤ P(A) ≤ 1

Frequentist interpretation:

• The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times.

Bayesian interpretation:

• A Bayesian interprets probability as a subjective degree of belief: For the same event, two separate people could have different viewpoints and so assign different probabilities.
• Largely popularized by revolutionary advance in computational technology and methods during the last twenty years.

Practice

Which of the following events would you be most surprised by?

(a) exactly 3 heads in 10 coin flips

(b) exactly 3 heads in 100 coin flips

(c) exactly 3 heads in 1000 coin flips

Practice

Which of the following events would you be most surprised by?

(a) exactly 3 heads in 10 coin flips

(b) exactly 3 heads in 100 coin flips

(c) exactly 3 heads in 1000 coin flips

Law of large numbers

Law of large numbers states that as more observations are collected, the proportion of occurrences with a particular outcome, p̂n, converges to the probability of that outcome, p.

Disjoint and non-disjoint outcomes

Non-disjoint outcomes: Can happen at the same time.

• A student can get an A in Stats and A in Econ in the same semester.

​

Disjoint (mutually exclusive) outcomes: Cannot happen at the same time.

• The outcome of a single coin toss cannot be a head and�a tail.
• A student both cannot fail and pass a class.
• A single card drawn from a deck cannot be an ace and�a queen.

Union of non-disjoint events

Figure from http://www.milefoot.com/math/discrete/counting/cardfreq.htm

What is the probability of drawing a jack or a red card from a well shuffled full deck?

Practice

What is the probability that a randomly sampled student thinks marijuana should be legalized or they agree with their parents' political views?

​

​

​

​

(a) (40 + 36 - 78) / 165

(b) (114 + 118 - 78) / 165

(c) 78 / 165

(d) 78 / 188

(e) 11 / 47

Practice

What is the probability that a randomly sampled student thinks marijuana should be legalized or they agree with their parents' political views?

​

​

​

​

(a) (40 + 36 - 78) / 165

(b) (114 + 118 - 78) / 165

(c) 78 / 165

(d) 78 / 188

(e) 11 / 47

Recap

General addition rule

P(A or B) = P(A) + P(B) - P(A and B)

​

​

​

​

Note: For disjoint events P(A and B) = 0, so the above formula simplifies to P(A or B) = P(A) + P(B)

Probability distributions

• The probability distribution for the genders of two kids:� Event MM FF MF FM� Probability 0.25 0.25 0.25 0.25

• Rules for probability distributions:

1. The events listed must be disjoint

2. Each probability must be between 0 and 1

3. The probabilities must total 1

A probability distribution lists all possible events and the probabilities with which they occur.

• The probability distribution for the gender of one kid:� Event Male Female� Probability 0.5 0.5

Practice

In a survey, 52% of respondents said they are Democrats. What is the probability that a randomly selected respondent from this sample is a Republican?

(a) 0.48

(b) more than 0.48

(c) less than 0.48

(d) cannot calculate using only the information given

Practice

In a survey, 52% of respondents said they are Democrats. What is the probability that a randomly selected respondent from this sample is a Republican?

(a) 0.48

(b) more than 0.48

(c) less than 0.48

(d) cannot calculate using only the information given

If the only two political parties are Republican and Democrat, then (a) is possible. However it is also possible that some people do not affiliate with a political party or affiliate with a party other than these two. Then (c) is also possible. However (b) is definitely not possible since it would result in the total probability for the sample space being above 1.

Practice

Complementary events are two mutually exclusive events whose probabilities that add up to 1.

• A couple has one kid. If we know that the kid is not a boy, what is gender of this kid? { M, F } Boy and girl are complementary outcomes.
• A couple has two kids, if we know that they are not both girls, what are the possible gender combinations for these kids?

Sample space is the collection of all possible outcomes of a trial.

• A couple has one kid, what is the sample space for the gender of this kid? S = {M, F}
• A couple has two kids, what is the sample space for the gender of these kids?

S = {MM, FF, FM, MF}

S = { MM, FF, FM, MF }

Independence

• Knowing that the coin landed on a head on the first toss does not provide any useful information for determining what the coin will land on in the second toss.�>> Outcomes of two tosses of a coin are independent.
• Knowing that the first card drawn from a deck is an ace does provide useful information for determining the probability of drawing an ace in the second draw.�>> Outcomes of two draws from a deck of cards (without replacement) are dependent.

Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other.

Practice

Between January 9-12, 2013, SurveyUSA interviewed a random sample of 500 NC residents asking them whether they think widespread gun ownership protects law abiding citizens from crime, or makes society more dangerous. 58% of all respondents said it protects citizens. 67% of White respondents, 28% of Black respondents, and 64% of Hispanic respondents shared this view. Which of the below is true?

​

Opinion on gun ownership and race ethnicity are most likely

(a) complementary

(b) mutually exclusive

(c) independent

(d) dependent

(e) disjoint

http://www.surveyusa.com/client/PollReport.aspx?g=a5f460ef-bba9-484b-8579-1101ea26421b

Practice

Between January 9-12, 2013, SurveyUSA interviewed a random sample of 500 NC residents asking them whether they think widespread gun ownership protects law abiding citizens from crime, or makes society more dangerous. 58% of all respondents said it protects citizens. 67% of White respondents, 28% of Black respondents, and 64% of Hispanic respondents shared this view. Which of the below is true?

​

Opinion on gun ownership and race ethnicity are most likely

(a) complementary

(b) mutually exclusive

(c) independent

(d) dependent

(e) disjoint

http://www.surveyusa.com/client/PollReport.aspx?g=a5f460ef-bba9-484b-8579-1101ea26421b

Checking for independence

If P(A occurs, given that B is true) = P(A | B) = P(A),�then A and B are independent.

P(randomly selected NC resident says gun ownership protects citizens, given that the resident is white)

= P(protects citizens | White) = 0.67

P(protects citizens | Black) = 0.28

P(protects citizens | Hispanic) = 0.64

P(protects citizens) = 0.58

​

P(protects citizens) varies by race/ethnicity, therefore opinion on gun ownership and race ethnicity are most likely dependent.

Determining dependence

based on sample data

We saw that P(protects citizens | White) = 0.67 and P(protects citizens | Hispanic) = 0.64. Under which condition would you be more convinced of a real difference between the proportions of Whites and Hispanics who think gun widespread gun ownership protects citizens?�n = 500 or n = 50,000

• If conditional probabilities calculated based on sample data suggest dependence between two variables, the next step is to conduct a hypothesis test to determine if the observed difference between the probabilities is likely or unlikely to have happened by chance.
• If the observed difference between the conditional probabilities is large, then there is stronger evidence that the difference is real.
• If a sample is large, then even a small difference can provide strong evidence of a real difference.

Determining dependence

based on sample data

We saw that P(protects citizens | White) = 0.67 and P(protects citizens | Hispanic) = 0.64. Under which condition would you be more convinced of a real difference between the proportions of Whites and Hispanics who think gun widespread gun ownership protects citizens?�n = 500 or n = 50,000

• If conditional probabilities calculated based on sample data suggest dependence between two variables, the next step is to conduct a hypothesis test to determine if the observed difference between the probabilities is likely or unlikely to have happened by chance.
• If the observed difference between the conditional probabilities is large, then there is stronger evidence that the difference is real.
• If a sample is large, then even a small difference can provide strong evidence of a real difference.

Product rule for independent events

P(T on the first toss) x P(T on the second toss)

= (1 / 2) x (1 / 2) = 1 / 4

You toss a coin twice, what is the probability of getting two tails�in a row?

P(A and B) = P(A) x P(B)

Or more generally, P(A1 and … and Ak) = P(A1) x … x P(Ak)

Practice

A recent Gallup poll suggests that 25.5% of Texans do not have health insurance as of June 2012. Assuming that the uninsured rate stayed constant, what is the probability that two randomly selected Texans are both uninsured?

(a) 25.5²

(b) 0.255²

(c) 0.255 x 2

(d) (1 - 0.255)²

​

​

​

​

​

http://www.gallup.com/poll/156851/uninsured-rate-stable-across-states-far-2012.aspx

​

​

Practice

A recent Gallup poll suggests that 25.5% of Texans do not have health insurance as of June 2012. Assuming that the uninsured rate stayed constant, what is the probability that two randomly selected Texans are both uninsured?

(a) 25.5²

(b) 0.255²

(c) 0.255 x 2

(d) (1 - 0.255)²

​

​

​

​

​

http://www.gallup.com/poll/156851/uninsured-rate-stable-across-states-far-2012.aspx

​

​

Disjoint vs. complementary

Do the sum of probabilities of two disjoint events always add up to 1?

Do the sum of probabilities of two complementary events always�add up to 1?

Yes, that's the definition of complementary, e.g. heads and tails.

Not necessarily, there may be more than 2 events in the sample space, e.g. party affiliation.

​

Putting everything together...

If we were to randomly select 5 Texans, what is the probability that at least one is uninsured?}

• If we were to randomly select 5 Texans, the sample space for the number of Texans who are uninsured would be:� S = {0, 1, 2, 3, 4, 5}
• We are interested in instances where at least one person is uninsured:� S = {0, 1, 2, 3, 4, 5}
• So we can divide up the sample space into two categories:� S = {0, at least one}

Putting everything together...

= 1 - (1 - 0.255)⁵

= 1 - 0.745⁵

= 1 - 0.23

= 0.77

​

At least 1:

P(at least one) = 1 - P(none)

Since the probability of the sample space must add up to 1:

P(at least 1 uninsured)

= 1 - P(none insured)

Practice

Roughly 20% of undergraduates at a university are vegetarian or vegan. What is the probability that, among a random sample of 3 undergraduates, at least one is vegetarian or vegan?

(a) 1 - 0.2 x 3

(b) 1 - 0.2³

(c) 0.8³

(d) 1 - 0.8 x 3

(e) 1 - 0.8³

Practice

P(at least 1 from veg)

= 1 - P(none veg)

= 1 - 0.8³

= 1 - 0.512 = 0.488

Roughly 20% of undergraduates at a university are vegetarian or vegan. What is the probability that, among a random sample of 3 undergraduates, at least one is vegetarian or vegan?

(a) 1 - 0.2 x 3

(b) 1 - 0.2³

(c) 0.8³

(d) 1 - 0.8 x 3

(e) 1 - 0.8³