11 Day 10

Announcements

You may have noticed I’ve been behind on grading corrections/getting lectures posted early…

I also have coursework, and for those who understand this image hopefully it’ll help convey why I’m less timely:

Exam 1 is October $2^{nd}$

We do not have the official room assignments yet, please do not ask me

• You will know when I know

There will be a full exam review prior

Exam test prep materials will be posted to the bookdown

• I might consider giving solutions

Homework 4 Corrections are posted

• If you want anything graded in the next 3 weeks submit it within the next 10 days

  • Otherwise, know in your heart of hearts that I am ignoring your emails

Review

Why do we study probability in statistics?

Probability

• A number between $0$ and $1$ that tells us how likely a given “event” is to occur

• Probability equal to $0$ means the event cannot occur

  • $P(x)=0$

• Probability equal to $1$ means the event must occur

  • $P(x)=1$

• Probability equal to $1/2$ means the event is as likely to occur as it is to not occur

  • $P(x)=0.5$

• Probability close to 0 (but not equal to 0) means the event is very unlikely to occur

  • The event may still occur, but we’d tend to be surprised if it did

• Probability close to 1 (but not equal to 1) means the event is very likely to occur

  • The event may not occur, but we’d tend to be surprised if it didn’t

• Experiment (in context of probability):

  • An activity that results in a definite outcome where the observed outcome is determined by chance

• Sample space:

  • The set of ALL possible outcomes of an experiment; denoted by $S$

1. Flip a coin once

$$S = \{H, T\}$$

2. Randomly select a person and then determine blood type

$$S = \{A, B, AB, O\}$$

• Event

  • A subset of outcomes belonging to sample space $S$

  • A capital letter towards the beginning of the alphabet is used to denote an event

• Simple event

  • An event containing a single outcome in the sample space $S$

• Example:

$$S = \{HH, HT, TH, TT\}$$

Let $A = "$we observe two heads$"$ $= \{HH\}$ is a simple event

• Compound event

  • An event formed by combining two or more events (thereby containing two or more outcomes in the sample space $S$)

• Example:

$$S = \{HH, HT, TH, TT\}$$

Let $B = "$we observe a head in the first or in the second flip$"$ $= \{HT, TH, HH\}$ is a compound event

• Subjective Probability

  • Probability is assigned based on judgement or experience

  • i.e. expert opinion, personal experience, “vibe math”

• Classical Probability

  • Make some assumptions in order to build a mathematical model from which we can derive probabilities

  • It’s not vibe math but it can definitely feel like it

$$P(A) = \frac{\text{number of outcomes in event } A}{\text{total number of outcomes in } S}.$$

• Relative or Empirical Probability

  • Think of the probability of an event as the proportion of times that the event occurs

$$P(x) \approx \frac{\text{number of times x is observed}}{\text{number of samples}}$$

• Law of Large Numbers

  • As the size of our sample (i.e., number of experiments) gets larger and larger:

  • The relative frequency of the event of our interest gets closer and closer to the true probability

Probability Continued

A probability model assigns a probability to each possible event constructed from the simple events in a particular sample space describing a particular experiment

• For a finite sample space with $n$ simple events, e.g. $S = \{E_1, E_2, \dots, E_n\}$:

  • The probability model assigns a number $p_i$ to event $E_i$ where $P(E_i) = p_i$ so that:

  1. $0 \leq p_i \leq 1$

  2. $p_1 + p_2 + \dots + p_n = 1$ (as a consequence, $P(S) = 1$)

• For an equally-likely probability model, the probability of observing $E_i$ is:

  $$P(E_i) = p_i = \frac{1}{n}$$

• If $A$ is an event in an equally-likely sample space $S$ and contains $k$ outcomes, then:

$$P(A) = \frac{\text{No. of outcomes in } A}{\text{No. of outcomes in } S} = \frac{k}{n}$$

Example

• Suppose there are 200 potential blood donors, and their blood types are classified & counted in the table below:

$$\begin{array}{|c|c|c|c|c|c|} \hline \textbf{Blood Type} & \textbf{A} & \textbf{B} & \textbf{AB} & \textbf{O} & \textbf{Total} \\ \hline \text{Count} & 80 & 20 & 10 & 90 & 200 \\ \hline \end{array}$$

• Our experiment involves selecting one donor and finding out what the blood type is

• We make a random selection so that everyone has the same chance of being selected

  • What kind of sample is this?

• The sample space $S$ is the set of all 200 donors

• The probability model for this experiment is given as follows:

$$\begin{array}{|c|c|c|c|c|c|} \hline \textbf{Outcome} & \textbf{A} & \textbf{B} & \textbf{AB} & \textbf{O} & \textbf{Total} \\ \hline \text{Count} & 0.4 & 0.1 & 0.05 & 0.45 & 1 \\ \hline \end{array}$$

• If $A$ is an event in $S$, then the event where $A$ does not occur is called the complement of $A$

• Denote the complement of $A$ by $A^c$ – read this as “A-complement”

Complement Rule

$$P(A^c) = 1 - P(A) \quad \text{or} \quad P(A) = 1 - P(A^c)$$

Note: This rule is useful when $P(A)$ is difficult to calculate but $P(A^c)$ is easy (or vice versa)

Example

• Suppose we roll a fair 6-sided die twice, then $S$ contains 36 equally-likely outcomes in the form of 36 ordered pairs, i.e. $(1, 1), (1, 2), \dots, (6, 5), (6, 6)$

• Let $A$ be “roll doubles”

  • Then $P(A) = \frac{6}{36} = \frac{1}{6}$

• $A^c$ is the event we “do not roll doubles”, and:

  $$P(A^c) = 1 - P(A) = 1 - \frac{1}{6} = \frac{5}{6}$$

Note: We could have counted the number of non-doubles in $S$, but this requires more effort

Unions and Intersections

From this point forward, there will be picture drawing

• The union of two events $A$ and $B$, denoted $A \cup B$, are all outcomes that belong to $A$, $B$, or both

  • Saying $A \cup B$ is equivalent to saying “A or B”

• The intersection of two events $A$ and $B$, denoted $A \cap B$, are all outcomes that belong to both $A$ and $B$

  • Saying $A \cap B$ is equivalent to saying “A and B”

Example In rolling a die once, consider events $A$ and $B$:

$A$: Roll an even number: $\{2, 4, 6\}$

$B$: Roll a number greater than 4: $\{5, 6\}$

$$\Rightarrow A \text{ or } B = \{2, 4, 5, 6\}$$

$$\Rightarrow A \text{ and } B = \{6\}$$

Example 1

In a statistics class of 30 students, there were 13 men and 17 women. Two of the men and three of the women received an A in this course. A student is chosen at random from the class

1. Find the probability that the student is a woman

2. Find the probability that the student received an A

3. Find the probability that the student is a woman or received an A

Mutual Exclusivity

• Two events $A$ and $B$ are mutually exclusive if they do not share any common outcomes

• Roll a die:

  • $A$: Roll a 1 or a 2: $\{1, 2\}$

  • $B$: Roll an even number: $\{2, 4, 6\}$

  • $C$: Roll a 3, 4, or 5: $\{3, 4, 5\}$

• Events $A$ and $C$ are mutually exclusive. Knowing that we rolled a 1 or 2 implies that we did not roll a 3, 4, or 5

• Events $A$ and $B$ are not mutually exclusive as $$A \text{ and } B = \{2\}$$

Addition rule for mutually exclusive events $A$ and $B$:

$$P(A \text{ or } B) = P(A) + P(B)$$

Conditional Probability

• A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred

• $P(A|B)$ denotes the conditional probability of event $A$ given that event $B$ has already occurred

$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$$

• Similarly,

$$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$$

Example 2

An economist predicts a 60% chance that stock $A$ will perform poorly and a 25% chance that stock $B$ will perform poorly. There is also a 16% chance that both stocks will perform poorly

What is the probability that stock $A$ performs poorly given that stock $B$ performs poorly?

Example 3

Below table shows the results from an experiment in which college students were given either 4 quarters or a $1 bill

$$\begin{array}{|c|c|c|} \hline \textbf{} & \textbf{Purchased Gum} & \textbf{Kept the Money} \\ \hline \text{Students Given Four Quarters} & 27 & 12 \\ \hline {\text{Students Given a 1 Dollar Bill}} & 19 & 33 \\ \hline \end{array}$$

Find the probability of randomly selecting a student who:

1. spent the money, given that the student was given 4 quarters (round to 3 decimal places)

2. kept the money, given that the student was given a $1 bill (round to 3 decimal places)

Multiplication Rule

From the definition of the conditional probability

$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)},$$

we have the following multiplication rule for finding $P(A \text{ and } B)$:

$$P(A \text{ and } B) = P(A|B) P(B)$$

Independence

• Events $A$ and $B$ are independent if the outcome of $A$ does not affect the outcome of $B$ and vice versa

• In terms of conditional probability, the probability of $A$ does not change given $B$ happened and vice versa

• That is, $A$ and $B$ are independent if one of the following is true:

  • $P(A|B) = P(A)$

  • $P(B|A) = P(B)$

  • $P(A \text{ and } B) = P(A)P(B)$

(You can show that all three statements are equivalent.)

Multiplication Rule for Independent Events

If $A$ and $B$ are independent, then:

$$P(A \text{ and } B) = P(A)P(B)$$

Example

Suppose we roll a fair die twice. What is the probability that the first roll is a 1 and the second roll is a 6?

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