Probabilities

These slides are based on slides developed by Mine Çetinkaya-Rundel

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

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

Practice

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

exactly 3 heads in 10 coin flips
exactly 3 heads in 100 coin flips
exactly 3 heads in 1000 coin flips

Practice

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

exactly 3 heads in 10 coin flips
exactly 3 heads in 100 coin flips
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, p, converges to the probability of that outcome, p.

Practice

When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss?

0.5
less than 0.5
more than 0.5

Practice

The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss.
P(H on 11th toss) = P(T on 11th toss) = 0.5
The coin is not “due” for a tail

Gambler’s Fallacy

The common misunderstanding of the law of large numbers is that random processes are supposed to compensate for whatever happened in the past; this is just not true and is also called law of averages.

Disjoint and non-disjoint outcomes

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

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

Union of non-disjoint events

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

There are 52 cards in a standard deck.
There are 4 suits; Diamonds, Clubs, Hearts, Spades.
Black cards include all Clubs and Spades.
Red cards include all Hearts and Diamonds.

Union of non-disjoint events

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

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(jack or red) = P(jack) + P(red) - P(jack and red)
P(jack) = 4/52
P(red) = 26/52
P(jack and red) = 2/52
28/52

Practice

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

	Shares Parent’s View	Doesn’t share	Total
NO legalize	40	11	51
YES legalize	78	36	114
Total	118	47	165

Practice

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

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(YES legalize or agree with parents) = P(YES legalize) + P(agree with parents) - P(YES legalize and agree with parents)

Practice

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

(40 + 36 - 78) / 165
(114 + 118 - 78) / 165
78 / 165
78 / 188
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?

(40 + 36 - 78) / 165
(114 + 118 - 78) / 165
78 / 165
78 / 188
11 / 47

Union of non-disjoint events

General rule:

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

In-class activity

Go to gradescope and answer the questions about the union of non-disjoint events

Probability Distributions

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

The probability distribution a coin toss:

	Head	Tail
Probability	0.50	0.50

Rules for probability distributions:

The events listed must be disjoint
Each probability must be between 0 and 1
The probabilities must total 1

Probability Distributions

Rules for probability distributions:

The events listed must be disjoint
Each probability must be between 0 and 1
The probabilities must total 1

The probability distribution two coin tossess:

	Head Head	Head Tail	Tail Head	Tail Tail
Prob.	0.25	0.25	0.25	0.25

Sample space

Sample space is the collection of all possible outcomes of an experiment

I flip a fair coin once, what is the sample space for the outcomes? S = {H, T}
I flip a fair coin twice, what is the sample space for the outcomes?
I flip a fair coin three times, what is the sample space for the outcomes?

Sample space

Sample space is the collection of all possible outcomes of an experiment

I flip a coin once, what is the sample space for the outcomes? S = {H, T}
I flip a coin once twice, what is the sample space for the outcomes? S = {HH, HT, TT, TH}
I flip a coin once three times, what is the sample space for the outcomes?

Sample space

Sample space is the collection of all possible outcomes of an experiment

I flip a coin once, what is the sample space for the outcomes? S = {H, T}
I flip a coin once twice, what is the sample space for the outcomes? S = {HH, HT, TT, TH}
I flip a coin once three times, what is the sample space for the outcomes? S = {HHH, HHT, HTH, HTT, THH, TTH, TTH, TTT}

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?

0.48
more than 0.48
less than 0.48
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?

0.48
more than 0.48
less than 0.48
cannot calculate using only the information given

Practice

If the only two political parties were Republican and Democrat, then 1 would be 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 3 is also possible. However 2 is definitely not possible since it would result in the total probability for the sample space being above 1.

Complementary events

Complementary events are two mutually exclusive events where one event consists of all outcomes that are not in the other event (the two probabilities add to 1).

Example: Rolling a Six-Sided Die

The sample space for rolling a standard six-sided die is: S = {1, 2, 3, 4, 5, 6}

I roll a die once. If we know that I didn’t roll an even number, what are the possible outcomes? {1, 3, 5}

Practice

If the probability of rain tomorrow is 0.3, what is the probability that it will NOT rain tomorrow?
In a standard deck of 52 playing cards, what is the probability of NOT drawing a diamond?
A fair die is rolled once. If event A is “rolling a number less than 4,” what is the probability of the complementary event A’?
In a class of 40 students, 25 students passed a math test. What is the probability that a randomly selected student did NOT pass the test?

Practice

If the probability of rain tomorrow is 0.3, what is the probability that it will NOT rain tomorrow? 0.7
In a standard deck of 52 playing cards, what is the probability of NOT drawing a diamond? 3/4
A fair die is rolled once. If event A is “rolling a number less than 4,” what is the probability of the complementary event A’? 1/2
In a class of 40 students, 25 students passed a math test. What is the probability that a randomly selected student did NOT pass the test? 15/40, 3/8

Independence

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

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.

Independence

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.

Independence

If probability that A occurs given that B is true equals the probability of A, then A and B are independent:

P(A occurs, given that B is true) = P(A | B) = P(A)

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.

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

Practice

P(protects citizens) = 0.58

P(protects citizens | White) = 0.67

P(protects citizens | Black) = 0.28

P(protects citizens | Hispanic) = 0.64

Practice

P(protects citizens) = 0.58

P(protects citizens | White) = 0.67 ≠ P(protects citizens)

P(protects citizens | Black) = 0.28 ≠ P(protects citizens)

P(protects citizens | Hispanic) = 0.64 ≠ P(protects citizens)

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

Product rule for independent events

The Product Rule for Independent Events states that if two events A and B are independent, then the probability of both events occurring (their intersection) is equal to the product of their individual probabilities.

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

P(A ∩ B) = P(A) × P(B)

Product rule for independent events

P(A ∩ B) = P(A) x P(B)

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

P(T₁ ∩ T₂) = P(T₁) × P(T₂) = 1/2 × 1/2 = 1/4

Product rule for independent events

The rule only applies when events are independent (the occurrence of one event doesn’t affect the probability of the other)
Independence must be verified before applying the product rule
This rule provides a simple way to calculate compound probabilities for independent events

The product rule is fundamental in probability theory and is used extensively in applications ranging from genetics to quality control in manufacturing.

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?

\(25.52\)
\(0.255^2\)
\(0.255 * 2\)
\((1 - 0.255)^2\)

Practice

\(0.255^2\)

Practice

A fair coin is flipped 3 times. What is the probability of getting heads on all three flips?
A bag contains 5 red marbles and 5 blue marbles. If you draw 2 marbles with replacement, what is the probability of drawing 2 blue marbles?
A six-sided die is rolled twice. What is the probability of rolling a 6 on the first roll AND an even number on the second roll?

Practice

A fair coin is flipped 3 times. What is the probability of getting heads on all three flips? \(\frac{1}{2}*\frac{1}{2}*\frac{1}{2} = \frac{1}{8}\)
A bag contains 5 red marbles and 5 blue marbles. If you draw 2 marbles with replacement, what is the probability of drawing 2 blue marbles? \(\frac{1}{2}*\frac{1}{2} = \frac{1}{4}\)
A six-sided die is rolled twice. What is the probability of rolling a 6 on the first roll AND an even number on the second roll? \(\frac{1}{6}*\frac{1}{2} = \frac{1}{12}\)

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?

Disjoint vs. complementary

Do the sum of probabilities of two disjoint events always add up to 1? Not necessarily, there may be more than 2 events in the sample space, e.g. party affiliation.
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