math day 13+14

randomness

Possible outcomes are known but it is uncertain which will occur for any given observation.

random phenomena

Everyday situations or processes in which the outcome is random (uncertain).

probability

• The way we quantify the uncertainty of randomness.

• It is often interpreted as the proportion of times an event occurs when the random phenomenon is repeated under similar conditions.

Long-run Probability

For random phenomena (random experiments), the outcome is uncertain.

In the short run, the proportion of times that something happens is

highly random

In the long run, the proportion of times that something happens becomes

predictable

Long-run probability:

The probability of a particular outcome is the proportion of times that the outcome would occur in the long-run,

i.e., long-run proportion.

Question: What is the probability that the home team wins the toss if they call heads?

• Clearly, we should all answer they have

50/50 shot – 50% chance – probability of 0.5

• In reality, however, the proportion on one toss will be

0/1 =0

or

1/1=1

law of large numbers

as a sample size grows larger, the average of the results gets closer to the true expected value of the population

experiment

the process of observing a phenomenon that has variation in its outcomes, i.e., any action or observable experiment where the outcome is random.

outcome (simple event)

Each individual result from the experiment.

• Notation: e1, e2, e3, ....

In an experiment, possible outcomes are ___ but it is ____

which will occur for any given observation.

known, uncertain

Example: Flipping a coin → e1=Heads; e2=Tails

Rolling a die → e1=1, e2=2, ..., e6=6

sample space

the collection of all possible outcomes of the experiment.

sample space for rolling a die

S = {1, 2, 3, 4, 5, 6}

sample space for flipping two coins

S = {HH, HT ,TH, TT}

event

the set of outcomes possessing a designated feature.

(i.e., the subset of the sample space.)

event example

Examples

Event A: we rolled a 4

A = {4}

Event B: flipped a heads

B = {H}

Event C: flipped heads and then tails

C = {HT}

Probability of an event A = P(A)

A numerical value representing the proportion of times the event A is expected to occur.

uniform probability model

In many situations, we assume that each elementary outcome is as clikely to occur as any other

P(A)=

Event Operation: complement

o The complement of an event A (A^c) is the set of all outcomes not in A.

o The occurrence of A^c means that A does not occur.

Disjoint (Mutually Exclusive) Events

o Two events are disjoint if they do not have any common outcomes.

• A = Rolling a multiple of three; B = Rolling a submultiple of four.

event operation: intersection

o The intersection of A and B (A ∩ B) consists of outcomes that are both in A and B.

o The occurrence of A ∩ B means that both A and B occur.

Event Operation: Union

o The union of A and B (A U B) consists of outcomes that are A or B or both.

o The occurrence of A (AUB) means that either A or B or both occur.

Complement rule:

The probability of something not happening is 1 minus the probability of it happening

Addition Rule:

The probability of A ∪ B happening is the probability of A, plus the probability

of B, minus the probability of A ∩ B. This is to avoid double counting the

probability that they both happen.

Addition Rule for disjoint events:

• For disjoint events A and B, we know that P(A and B) = P A ∩ B = 0

• Therefore, the additive rule becomes: P(A or B) = P A ∪ B = P A + P B

Independent Events

Two events are independent if the fact that one event has occurred doesn’t affect the probability that the other event will occur

dependent events

If whether or not one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent

We can assume independence:

• Repeated sampling from any population where individuals are “replaced”

• Repeated sampling from very large populations where individuals are NOT “replaced”

Independent Events vs. Disjoint Events

The idea of independent events is about whether or not the events affect each other in the sense that the occurrence of one event affects the probability of the occurrence of the other.

• The idea of disjoint events is about whether or not it is possible for the events to occur at the same time.

Multiplication Rule for Independent Events

If A and B are independent, then

Probability of A ∩ B = probability of A × Probability of B

P(A ∩ B) = P(A) × P(B)

conditional property

The conditional probability of event A, given that event B has occurred is…

It is the proportion of the population having the characteristic A among all

those having the characteristic B.

denominator for conditional property

The probability in the denominator is always for the “given” event.

Multiplication Rule for Dependent Events

If A and B are dependent, then

Probability of A ∩ B = Probability of A × Probability of B given A

= Probability of B × Probability of A given B

P( A ∩ B ) = P( A ) × P( B | A )

= P( B ) × P( A | B )

Checking for Independence

o If two events are independent, the occurrence of one event does not affect the probability of the other.

o Two events A and B are independent if any one of the following equivalent conditions holds:

• P( A | B ) = P A ;

• P (B | A ) = P B ;

• P A ∩ B = P (A) P ( B)

o Only one condition needs to be verified. Choose the condition that is most convenient based on the given probability information.

A couple plans to have four children. The father notes that the sample space for the number of girls the couple can have is​ 0, 1,​ 2, 3, and 4. He goes on to say that since there are five outcomes in the sample​ space, and since each child is equally likely to be a boy or​ girl, all five outcomes must be equally likely.​ Therefore, the probability of all four children being girls is​ 1/5. Explain the flaw in his reasoning.

The outcomes are not equally likely.

When we group these 16 sequences by the total number of girls, some counts can happen in many more ways than others:

• 0 Girls: Only 1 way (BBBB) 1/16 chance

• 1 Girl: 4 ways (GBBB, BGBB, BBGB, BBBG) 4/16 chance

• 2 Girls: 6 ways (GGBB, GBGB, GBBG, BGGB, BGBG, BBGG) 6/16 chance

• 3 Girls: 4 ways (GGGB, GGBG, GBGG, BGGG) 4/16 chance

• 4 Girls: Only 1 way (GGGG) 1/16 chance

A patient is told that a test for a certain disease is​ 92% accurate.

Let D denote​ {person has the​ disease}, and let P denote​ {person tests​ positive}. Using these events and their​ complements, express as a conditional probability the event that a person without the disease tests negative.

$$P(P^c \mid D^c) \quad \text{or} \quad P(\bar{P} \mid \bar{D})$$

For [Group X], [Outcome Y] happens," Group X is the condition (the given) and Outcome Y is what we are measuring.

Who are we starting with? and what are we observing

Law of Total Probability

Bayes’ Theorem

Once event A has occurred, the updated or posterior probability of B is given by the conditional probability:

they are mutually exclusive and dependent, since one occurring prevents the other one from occurring.,

can you only be mutually exclusive and dependent

For events with non-zero probabilities, yes—if two events are mutually exclusive, they MUST be dependent.

They cannot be mutually exclusive and independent at the same time (unless one of the events has a 0% chance of happening).

Suppose the following tree diagram summarizes the responses of 500 people to two​ questions, where the first response is either yes or no and the second is multiple choice with three possible answers​ (A, B or​ C). Use the tree diagram to calculate the probability that a person answered B to the multiple choice question.

.31

according to a​ survey, 65.7% of all adults between the ages of 18 and 44 were considered current drinkers. Based on this​ estimate, if two randomly selected adults between the ages of 18 and 44 are​ selected, what is the probability that at least one is a current​ drinker?

The probability that at least one of the two randomly selected adults is a current drinker is .88

Find the Probability of the Complement Event: Probability an adult IS NOT a current drinker:

$$P(D^c) = 1 - 0.657 = 0.343$$

Step 2: Calculate P(neither is a drinker})

Assuming the selection of the two adults is independent:

$$P(\text{neither is a drinker}) = P(D^c) \times P(D^c) = (0.343)^2 = 0.117649$$

Step 3: Calculate P(at least one drinker})

$$P(\text{at least one drinker}) = 1 - 0.117649 = \mathbf{0.882351}$$

The Exponent Rule (For Repeated Identical Actions) for finding sample spaces

$$\text{Total Outcomes} = (\text{Outcomes per single trial})^k$$

Flipping 10 coins:

• Single trial outcomes: 2 (Heads, Tails)

• Number of flips (k)= 10

• Formula: 2^{10} = 1,024 outcomes