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Introduction to Probability

Zhaoxia Yu

Outline

  • Experiment and Sample Space
  • Event
  • Probability
  • Random Variable
  • Distribution of a Random Variable

Introduction

  • We have used plots and summary statistics to learn about the distribution of different variables in the observed data and to investigate their relationships.

  • We now want to generalize our findings to the entire population of interest.

  • However, we almost always remain uncertain about the true distributions and relationships in the population.

  • Therefore, when we generalize our findings from a sample to the whole population, we should explicitly specify the extent of our uncertainty.

  • We use probability to measure or quantify uncertainty.

Experiment and Sample Space

Experiment

  • Experiment: any process that produces one outcome from a set of possible outcomes.

  • Before performing the experiment, we know the possible outcomes but do not know which one will occur.

Experiment: Examples

  • roll a die.

  • flip a coin 10 times.

  • measure the resting heart rate of a randomly selected participant.

  • randomly select a patient and record their blood type.

Sample Space

Sample space: the set of all possible outcomes of an experiment. Denoted by the symbol \(\mathbf S\). Examples:

  • For rolling a die, the sample space is \(\mathbf S = \{1, 2, 3, 4, 5, 6\}\).

  • For flipping a coin 10 times, the sample space is

    \[\begin{aligned} \mathbf S = \{& HHHHHHHHHH,\; HHHHHHHHHT,\; \ldots,\\ & TTTTTTTTTH,\; TTTTTTTTTT \}. \end{aligned}\]

Sample Space: Examples (cont.)

  • For measuring the resting heart rate of a randomly selected participant, the sample space is \(\mathbf S = \{0, 1, 2, \ldots\}\) (assuming heart rate is measured in beats per minute).

  • For randomly selecting a patient and recording their blood type, the sample space is \(\mathbf S = \{A, B, AB, O\}\).

Experiment and Sample Space

  • An experiment is the process we perform.

  • Once the experiment is specified, the sample space is determined. It contains all possible outcomes of the experiment.

  • From the sample space, we can define different events and random variables depending on the question we want to answer.

Events

Event: Definition

  • An event is a subset of the sample space.

  • An event occurs if the observed outcome belongs to that subset.

Example: Roll a Die

Experiment: - Roll a fair die.

Sample space: \(\mathbf S=\{1,2,3,4,5,6\}\)

Events:

  • \(A=\{2,4,6\}\): roll an even number

  • \(B=\{5,6\}\): roll a number greater than 4

  • \(C=\{1\}\): roll a 1

Two Special Events

Every sample space contains two special events.

  • Impossible event: \(\emptyset\) Contains no outcomes.

  • Certain event: \(\mathbf S\)

    Contains every possible outcome.

Example

For \(\mathbf S=\{1,2,3,4,5,6\}\),

  • “Roll a 7” =\(\emptyset\).

  • “Roll a number between 1 and 6” = \(\mathbf S\).

Set Operations on Events

Suppose

\(A=\{2,4,6\},\qquad B=\{4,5,6\}.\)

Then

  • Union: \(A\cup B=\{2,4,5,6\}\)

  • Intersection: \(A\cap B=\{4,6\}\)

  • Complement: \(A^c=\{1,3,5\}\)

Relationships Between Events

Suppose

\[A=\{2,4,6\},\qquad B=\{4,6\},\qquad C=\{1,3,5\}.\]

  • \(B\subset A\)

  • \(A\cap C=\emptyset\)

  • \(A\) and \(C\) are disjoint (mutually exclusive)

  • \(A\cup C=\mathbf S\)

  • \(C=A^c\)

Probability

Probability

  • A probability function assigns a number to each event.

  • The probability of an event measures how likely the event is to occur.

  • We write the probability of an event \(A\) as

\[ P(A). \]

  • A larger probability means the event is more likely to occur.

Example: A Fair Die

Suppose a fair six-sided die is rolled once.

Sample space:

\[ \mathbf S=\{1,2,3,4,5,6\}. \]

Since the die is fair, each outcome is equally likely.

\[ P(\{1\})=P(\{2\})=\cdots=P(\{6\})=\frac16. \]

Each outcome has probability \(\frac16\).

Probability of Events

Let

\[ A=\{2,4,6\}, \]

the event that the die shows an even number.

Since \(A=\{2\}\cup\{4\}\cup\{6\},\)

\[ P(A) = P(\{2\})+P(\{4\})+P(\{6\}) = \frac36 = \frac12. \]

The probability of an event is the sum of the probabilities of its outcomes.

Properties of a Probability Function

A probability function satisfies three basic properties.

  1. Nonnegative \(P(A)\ge0.\)

  2. Certain event \(P(\mathbf S)=1.\)

  3. Additivity. If \(A\) and \(B\) are disjoint,

\[ P(A\cup B)=P(A)+P(B). \]

Equally Likely Outcomes

When all outcomes are equally likely,

\[ P(A) = \frac{\text{number of outcomes in }A} {\text{number of outcomes in }\mathbf S}. \]

Example

For a fair die,

\[ P(\{2,4,6\})=\frac36=\frac12. \]

This shortcut applies only when all outcomes are equally likely.

Joint Probability

The joint probability of \(A\) and \(B\) is

\[ P(A\cap B), \]

the probability that both events occur.

Suppose a fair die is rolled once. Let

  • \(A=\{2,4,6\}\), the event that the outcome is even, and

  • \(B=\{4,5,6\}\), the event that the outcome is greater than 3.

Since \[ A\cap B=\{4,6\}, \]

\[ P(A\cap B)=\frac26=\frac13. \]

Marginal vs Joint Probability

The probability of a single event is called its marginal probability.

  • \(P(A)\) is the marginal probability of \(A\).

  • \(P(B)\) is the marginal probability of \(B\).

  • \(P(A\cap B)\) is the joint probability of \(A\) and \(B\).

Marginal probabilities describe one event at a time, while a joint probability describes two events occurring together.

Conditional Probability

Suppose we know that event \(B\) has occurred. What is the probability that event \(A\) also occurs? This is called the conditional probability of \(A\) given \(B\), denoted by

\[ P(A\mid B). \]

Definition: If \(P(B)>0\), the conditional probability of \(A\) given \(B\) is

\[ P(A\mid B)=\frac{P(A\cap B)}{P(B)}. \]

  • The denominator \(P(B)\) represents the probability of the condition.
  • The numerator \(P(A\cap B)\) is the probability that both \(A\) and \(B\) occur.
  • Conditional probability restricts our attention to outcomes in event \(B\).

Consider rolling a fair die once. Let \[ A=A=\{2,4,6\}, B=\{4,5,6\}\]

By the definition of \(P(A\mid B)\), we have \[P(A\mid B) = \frac{P(A\cap B)}{P(B)} = \frac{2/6}{3/6} = \frac23. \]

Conditional probability updates probabilities using additional information.

Independence

Two events are independent if knowing one event occurs does not change the probability of the other.

Equivalently,

\[ P(A\mid B)=P(A). \]

or

\[ P(A\cap B)=P(A)P(B). \]

In our example,

\[ P(A)=\frac12, \quad P(A\mid B)=\frac23. \]

Since

\[ P(A\mid B)\ne P(A), \]

the events \(A\) and \(B\) are dependent.

Independent Events

Let \(A=\{1,3,5\}, \quad B=\{1,2,3,4\}\)

Then, \[P(A)=\frac36=\frac12, P(B)=\frac46=\frac23, P(A\cap B)=\frac26=\frac13.\]

Since \[P(A\cap B) = P(A)P(B) = \frac12\times\frac23 = \frac13,\]

the events are independent.

Independent vs. Disjoint Events

Disjoint Independent
Cannot occur together Can occur together
\[A\cap B=\emptyset\] \[P(A\cap B)=P(A)P(B)\]
Knowing one occurs tells you the other cannot occur Knowing one occurs does not change the probability of the other

Example (Roll a Fair Die)

Let \(A=\{2,4,6\}, \quad B=\{1,3,5\}.\)

  • \(A\) and \(B\) are disjoint because \(A\cap B=\emptyset.\)

  • They are not independent because

\[ P(A\mid B)=0\ne\frac12=P(A). \]

Key Idea:
Disjoint events are mutually exclusive.
Independent events do not influence each other.

Caes Study

Case Study: Blood Type and Rh Factor

Suppose we randomly select one person from a population.

For this person, we observe

  • ABO blood type: A, B, AB, or O
  • Rh factor: Positive (+) or Negative (−)

Experiment and Sample Space

Experiment

Randomly select one person and determine the person’s blood type and Rh factor.

Sample Space

\[ \mathbf S= \{ \text{A+},\text{A−}, \text{B+},\text{B−}, \text{AB+},\text{AB−}, \text{O+},\text{O−} \}. \]

The sample space contains eight possible outcomes.

Events

Examples of events include

  • \(A=\{\text{A+},\text{A−}\}\)

    The person has blood type A.

  • \(B=\{\text{A+},\text{B+},\text{AB+},\text{O+}\}\)

    The person is Rh positive.

  • \(C=\{\text{O+},\text{O−}\}\)

    The person has blood type O.

Each event is a subset of the sample space.

Operations on Events

Using the previous events,

  • Union: \(A\cup C\)

    The person has blood type A or O.

  • Intersection: \(A\cap B\)

    The person has blood type A and is Rh positive.

  • Complement: \(B^c\)

    The person is Rh negative.

Relationships Between Events

Examples:

  • \(A\cap C=\emptyset\)

    A person cannot have both blood type A and blood type O.

  • Therefore, \(A\) and \(C\) are disjoint.

  • \(A\subset A\cup C\)

  • \(A\cup A^c=\mathbf S\)

These relationships hold for any events.

Probability

Suppose the blood type distribution in a population is

Blood Type Probability
A 0.42
B 0.10
AB 0.04
O 0.44

This is marginal dsitribution of blood type.

Recall

  • \(A\): person has blood type A

  • \(C\): person has blood type O

Then

  • \(P(A)=0.42\)

  • \(P(C)=0.44\)

Since \(A\) and \(C\) are disjoint,

\[ P(A\cup C) = P(A)+P(C) = 0.42+0.44 = 0.86. \]

ABO and Rh Factor

  • Suppose we also know he Rh factor distribution is

    \[P(\text{Rh+})=0.85,\qquad P(\text{Rh−})=0.15.\]

  • The ABO and Rh blood group systems are determined by different genes and are generally considered independent.

  • Therefore, the probability of having a particular ABO blood type does not depend on Rh status.

Joint and Conditional Probability

Since ABO blood type and Rh factor are independent,

\[ P(\text{A and Rh+}) = P(\text{A})P(\text{Rh+}) = 0.42\times0.85 = 0.357. \]

The probability that a randomly selected person has blood type A and is Rh positive is 0.357.

Using conditional probability,

\[ P(\text{A}\mid\text{Rh+}) = \frac{P(\text{A and Rh+})} {P(\text{Rh+})} = \frac{0.357}{0.85} = 0.42. \]

Knowing that a person is Rh positive does not change the probability that the person has blood type A.

Discrete Random Variable

A Motivating Example

Experiment: flip a fair coin 10 times.

Assumptions

  • The coin is fair.
  • The 10 flips are independent.

Question: What can we say about the outcomes?

Sample Space

Each outcome is a sequence of 10 heads (H) and tails (T).

Examples:

HHHTHTTTHH

HTHHHTTTHT

TTTTTHHHHH

There are \(2^{10}=1024\) possible sequences. Since the coin is fair and the flips are independent,

\[ P(\text{each sequence})=\frac1{2^{10}}=\frac1{1024}. \]

Events

An event is a subset of the sample space.

Examples of events:

  • \(A\): Exactly 4 heads

  • \(B\): At least 8 heads

  • \(C\): The first flip is a head

Each event contains many possible sequences.

Question: How many sequences belong to event \(A\)?

Do We Really Need All 1024 Sequences?

The sample space contains \(1024\) possible outcomes.

However, in many applications we do not care about the exact sequence. Instead, we ask questions like

  • How many heads were observed?

  • Was the first flip a head?

  • Were there at least 8 heads?

These questions summarize the outcome rather than describing the entire sequence.

Random Variables

A random variable assigns a numerical value to each outcome in the sample space.

Define \(X=\text{number of heads in 10 flips}.\)

Examples

Outcome \(X\)
HHHHHHHHHH 10
HHHTHTTTHH 6
TTTTTTTTTT 0
HTHTHTHTHT 5

Why Random Variables Help

Instead of studying \(1024\) possible sequences, we study only \(11\) possible values of \(X.\)

where \(X\in\{0,1,2,\ldots,10\}.\)

Random variables simplify probability problems.

Probability Distribution

Now our goal is to find

\[ P(X=x), \]

for each possible value \(x=0,1,\ldots,10.\)

For example,

\[ P(X=4) \]

is the probability of observing exactly four heads.

Example

To compute \(P(X=4),\) count the number of sequences with exactly four heads.

There are \(\binom{10}{4}=210\) such sequences. Therefore,

\[ P(X=4) = \frac{\binom{10}{4}}{2^{10}} = \frac{210}{1024} \approx0.205. \]

Another Example

Similarly,

\[ P(X=3) = \frac{\binom{10}{3}}{2^{10}}, \]

and

\[ P(X=7) = \frac{\binom{10}{7}}{2^{10}}. \]

Notice the pattern.

A General Formula

For any

\[ x=0,1,\ldots,10, \]

\[ P(X=x) = \frac{\binom{10}{x}}{2^{10}}. \]

More generally, if the probability of heads is \(p,\) then

\[ P(X=x) = \binom{10}{x} p^x (1-p)^{10-x}. \]

Binomial Distribution

A random variable follows a Binomial distribution if

  • there are a fixed number of trials,
  • each trial has two possible outcomes,
  • the trials are independent,
  • the probability of success is the same for every trial.

Notation:

\[ X\sim\text{Binomial}(n,p). \]

Its probability mass function is

\[ P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}, \]

for \(x=0,1,\ldots,n.\)

Does Our Example Follow a Binomial Distribution?

For the coin-flipping experiment,

  • Number of trials: \(n=10\)

  • Success: Head

  • Probability of success: \(p=\frac12\)

Therefore,

\[ X\sim\text{Binomial}\left(10,\frac12\right). \]

A Special Case: One Trial

The Binomial distribution describes the number of successes in \(n\) independent trials.

When $n=14, for example,

  • Flip a coin once.
  • Test whether a patient responds to a treatment.
  • Determine whether a person has blood type O.

Each experiment has only two possible outcomes:

  • Success
  • Failure

Bernoulli Distribution

A Bernoulli random variable records the outcome of a single trial.

Define

\[ X= \begin{cases} 1, & \text{if success},\\ 0, & \text{if failure.} \end{cases} \]

Denote \(p=P(X=1)\), then

\[ P(X=0)=1-p. \]

We write

\[ X\sim\text{Bernoulli}(p). \]

Bernoulli Distribution

The probability mass function (pmf) is

\[ P(X=x) = p^x(1-p)^{1-x}, \qquad x=0,1. \]

Equivalently,

\(x\) Probability
0 \(1-p\)
1 \(p\)

Example

Randomly select one person.

Define

\[ X= \begin{cases} 1, & \text{if the person has blood type O},\\ 0, & \text{otherwise.} \end{cases} \]

Then

\[ X\sim\text{Bernoulli}(p), \]

where

\[ p=P(\text{blood type O}). \]

The value of \(p\) depends on the population being studied.

From Experiment to Random Variable

  • Experiment: The process we perform.

  • Sample Space (\(\mathbf{S}\)): All possible outcomes of the experiment.

  • Events: Subsets of the sample space.

  • Random Variables: Numerical functions defined on the sample space.

Code 
flowchart TD
    A(Experiment)
    B(Sample Space)
    C(Events)
    D(Random Variables)

    A --> B
    B --> C
    B --> D

flowchart TD
    A(Experiment)
    B(Sample Space)
    C(Events)
    D(Random Variables)

    A --> B
    B --> C
    B --> D

Discrete vs. Continuous Random Variables

Discrete Random Variables

  • Take a countable set of possible values.
  • Typically represent counts or categories.
  • Examples:
    • Number of heads in 10 coin flips (Binomial)
    • Whether a person has blood type O (Bernoulli)
    • Number of patients visiting an emergency room in one day

Continuous Random Variables

  • Take any value within an interval of the real line.
  • Used to measure quantities rather than count them.
  • Examples:
    • Height
    • Weight
    • Body Mass Index (BMI)
    • Blood pressure

For a continuous random variable, between any two possible values, there are infinitely many other possible values.

Probability distributions

  • The probability distribution of a random variable provides the required information to find the probability of its possible values.

  • The probability distributions discussed here are characterized by one or more parameters.

  • The parameters of probability distributions we assume for random variables are usually unknown.

Discrete probability distributions

  • For discrete random variables, the probability distribution is fully defined by its probability mass function (pmf).

  • This is a function that specifies the probability of each possible value within the range of random variable.

  • For the genotype example, the pmf of the random variable \(X\) is

\[P(X=x) = \left\{ \begin{array}{l@{\quad}l} 0.49 & \mbox{for } x=0, \\ 0.42 & \mbox{for } x=1, \\ 0.09 & \mbox{for } x=2. \end{array} \right.\]

Discrete uniform distribution

  • One of the most straightforward, yet useful distributions, is the one that assigns the same probability, regardless of the value.

  • For that reason, it is called uniform probability distribution.

  • A discrete uniform distribution can be written as follow

\[P(X=x) = \left\{ \begin{array}{l@{\quad}l} \frac{1}{k} & x=1,...,k \\ 0 & \mbox{otherwise} \\ \end{array} \right.\]

Continuous probability distributions

  • For discrete random variables, the pmf provides the probability of each possible value.

  • For continuous random variables, the number of possible values is uncountable, and the probability of any specific value is zero.

  • For these variables, we are usually interested in the probability that the value of the random variable is within a specific interval from \(x_{1}\) to \(x_{2}\); we show this probability as \(P(x_{1} < X \le x_{2})\).

Probability density function

  • For continuous random variables, we use probability density functions (pdf) to specify the distribution.

  • Using the pdf, we can obtain the probability of any interval.

  • Example: (continuous) uniform distribution.

Continuous Uniform Distribution

  • We previously saw discrete uniform distribution; we now discuss its continuous version.

  • Let \(a\) and \(b\) be two real numbers, so that \(a<b\). The following the pdf of a continuous uniform distribution:

\[f(x) = \left\{ \begin{array}{l@{\quad}l} \frac{1}{b-a} & a<x<b \\ 0 & \mbox{otherwise} \\ \end{array} \right.\]

PDf of a Uniform Distribution

  • Lex \(X\sim U(a,b)\) where \(a<b\). Then its pdf \(f(x)\) is given by
Code 
a <- 2
b <- 8
h <- 1/(b-a)

# Set up empty plot
plot(0, 0,
     type = "n",
     xlim = c(0, 10),
     ylim = c(-0.03, h + 0.05),
     axes = FALSE,
     xlab = "",
     ylab = "")

# x-axis with arrow
arrows(0, 0, 10, 0, length = 0.08)

# Rectangle
rect(a, 0, b, h)

# Dashed vertical sides
segments(a, 0, a, h, lty = 2)
segments(b, 0, b, h, lty = 2)

# Labels
text(a, -0.012, "a")
text(b, -0.012, "b")
text(a - 0.55, h/2, expression(frac(1, b-a)))

Example: Uniform Distribution \(U(3,\,3.5)\)

  • Suppose Adam arrives at a coffee shop at a random time between 3:00 pm and 3:30 pm.

  • Every arrival time in this interval is equally likely.

  • Let \(X\) denote Adam’s arrival time. Then \[X \sim U(3,\,3.5).\]

PDF of \(U(3,\,3.5)\)

\[ f(x)= \begin{cases} \dfrac{1}{3.5-3}=2, & 3<x<3.5,\\[1ex] 0, & \text{otherwise.} \end{cases} \]

Code 
a <- 3
b <- 3.5
h <- 1/(b-a)

plot(0, 0,
     type = "n",
     xlim = c(2.8, 3.7),
     ylim = c(-0.15, 2.3),
     axes = FALSE,
     xlab = "", ylab = "")

arrows(2.8, 0, 3.7, 0, length = 0.08)

rect(a, 0, b, h)

segments(a, 0, a, h, lty = 2)
segments(b, 0, b, h, lty = 2)

text(a, -0.08, "3")
text(b, -0.08, "3.5")
text(a-0.08, h/2, "2")

Probability Is the Area Under the PDF

The probability of an interval is the area under the PDF.

For example,

\[ P(3.1<X<3.3) = 2\times(3.3-3.1) = 0.40. \]

  • the height of the PDF is 2,
  • the probability is the area between 3.1 and 3.3, which is a rectangle with width \(3.3-3.1=0.20\) and height \(2\). So the area is: \(2\times0.20=0.40.\)

Probability and Area Under the PDF

For \(X\si mU(3, 3.5)\), similarly, we can find the probability of any interval by calculating the area under the PDF.

  • \(P(3.09<X<3.11)=2\times(3.11-3.09)=0.04\)

  • \(P(3.099<X<3.101)=2\times(3.101-3.099)=0.004\)

  • Can you guess what \(P(X=3.1)\) is?

A PDF Is Not a Probability Function

  • In the \(X\sim U(3,\,3.5)\) example, the pdf is \(f(x)=2\) for \(3<x<3.5\).

  • From the previous slide, \(P(X=3.1)=0\).

  • So \(fx\) cannot be the probability that \(X=x\).

A PDF gives the density of probability, not the probability itself.

Key idea:

  • For a discrete random variable, probability is assigned to individual values.

  • For a continuous random variable, probability is assigned to intervals.

Probability Density Function

A probability density function (pdf) has two key properties:

  • The total area under the density curve is 1.

  • The probability that a continuous random variable falls within an interval is the area under the density curve over that interval.

Lower tail probability

  • the probability of observing values less than or equal to a specific value \(x\), is called the lower tail probability and is denoted as \(P(X \le x)\).

Upper tail probability

  • The probability of observing values greater than \(x\), \(P(X > x)\), is called the upper tail probability and is found by measuring the area under the curve to the right of \(x\).

Probability of intervals

  • The probability of any interval from \(x_{1}\) to \(x_{2}\), where \(x_{1} < x_{2}\), can be obtained using the corresponding lower tail probabilities for these two points as follows: \[\begin{equation*} P(x_1 < X \le x_2) = P(X \le x_2) - P(X \le x_1). \end{equation*}\]

  • For example, the probability of a BMI between 25 and 30 is \[\begin{equation*} P(25 < X \le 30) = P(X \le 30) - P(X \le 25). \end{equation*}\]

Normal distribution

  • Consider the probability distribution function and its corresponding probability density curve we assumed for BMI in the above example.

  • This distribution is known as normal distribution, which is one of the most widely used distributions for continuous random variables.

  • Random variables with this distribution (or very close to it) occur often in nature.

Normal distribution

  • A normal distribution and its corresponding pdf are fully specified by the mean (usually denoted as \(\mu\)) and variance (usually denoted as \(\sigma^2\)):

\[N(\mu, \sigma^2)\]

  • \(N(0, 1)\) is called the standard normal distribution.

Normal Distributions

The 68-95-99.7% rule

  • The 68–95–99.7% rule for normal distributions specifies that

    • 68% of values fall within 1 standard deviation of the mean

    • 95% of values fall within 2 standard deviations of the mean

    • 99.7% of values fall within 3 standard deviations of the mean

Normal Distribution

Normal Distribution

The 68-95-99.7% rule

Practice

  • Suppose \(Z\sim N(0,1)\).
  • \(P(Z>1)=0.135+0.0235+0.0015=0.16\).
  • \(P(Z<1)=1-P(Z>1)=1-0.16=0.84\)
  • \(P(Z<-1)=0.135+0.0235+0.0015=0.16\).
  • \(P(-1 < Z < 1)=P(Z<1)-P(Z<-1)=0.68\)

The Standard Normal Distribution

  • If \(X\sim N(\mu, \sigma)\), we can standardize in the following way \[Z=\frac{X-\mu}{\sigma},\] where \(Z\sim N(0,1)\).

Student’s t-distribution

Another continuous probability distribution that is used very often in statistics is the Student’s \(t\)-distribution or simply the \(t\)-distribution.

Additional Example

Going Further: Genotype vs. Blood Type

The ABO blood type is determined by a person’s genotype.

Each person inherits one allele from the mother, and one allele from the father.

There are three possible alleles: A, B, and O.

The six possible genotypes are

Genotype Blood Type
AA A
AO A
BB B
BO B
AB AB
OO O

Observation: Blood type A can arise from two different genotypes (AA or AO).

A Compatible Genotype Distribution

Suppose the genotype distribution is

Genotype Probability
AA 0.12
AO 0.30
BB 0.03
BO 0.07
AB 0.04
OO 0.44

The genotype distribution gives the blood type distribution

Blood Type Probability
A 0.42
B 0.10
AB 0.04
O 0.44

because

  • \(P(\text{A})=P(\text{AA})+P(\text{AO})=0.42\)

  • \(P(\text{B})=P(\text{BB})+P(\text{BO})=0.10\)

Conditional Probability

Suppose a randomly selected person has blood type A.

What is the probability that the genotype is AO?

Using conditional probability,

\[ P(\text{AO}\mid\text{A}) = \frac{P(\text{AO})}{P(\text{A})} = \frac{0.30}{0.42} = 0.714. \]

Similarly,

\[ P(\text{AA}\mid\text{A}) = \frac{0.12}{0.42} = 0.286. \]

Why Does This Matter?

Suppose two people both have blood type A.

  • Person 1 may have genotype AA.
  • Person 2 may have genotype AO.

Although they have the same blood type,

their children may inherit different blood types.

Genotype contains more information than blood type.

Take-Home Message

This case study illustrates several important ideas.

  • An experiment defines a sample space.

  • Events are subsets of the sample space.

  • Probabilities quantify uncertainty.

  • Conditional probability incorporates additional information.

  • Sometimes what we observe (blood type) does not uniquely determine the underlying state (genotype).

This is a common theme in data science and biomedical research.