10 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

\[S = \{H, T\}\]

  1. 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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