Textbook Notes (381,222)
CA (168,408)
UTSC (19,325)
Statistics (135)
STAB22H3 (130)
Moras (15)
Chapter 4

# Chapter 4

10 Pages
186 Views

Department
Statistics
Course Code
STAB22H3
Professor
Moras

This preview shows pages 1-3. Sign up to view the full 10 pages of the document.
Chapter 4
4.1 Randomness
Probability - a pattern that emerges clearly only after many repetitions.
The proportion of times the outcome will occur in a long series of repetitions.
4.1 Toss a coin 5000 times.
Probability describes only what happens in the long run. (repetitive trials)
The language of probability
Random in statistics is a kind of order that emerges in the long run.
Randomness and probability
Random â€“ when the individual outcomes are uncertain but there is a regular distribution of
outcomes in a large number of repetitions
Probability â€“ (random) â€“ repetitive trials (look at overall picture)
Fair â€“ has equally likely chances
Ex: Fair Coin = head and tail sides
-Long series of independent trials
-Observing many trials (done practically)
-Simulations useful when a lot trials are down without being practical
The uses of probability
Games of chance
Ex: dice, cards, or spinning a roulette wheel, gambling
Job wise
Ex: astronomy,
4.2 Probability Models
Probability model â€“ description of random phenomenon in the language of mathematics
Description of probability outcomes (probability model):
-A list of possible outcomes
-A probability for each outcome
Sample Space
The sample Space (S) of a random phenomenon is the set of all possible outcomes.
-Each possible outcome is a sample and the sample space contains all possible samples.
www.notesolution.com
Ex 4.6 Sample space for tossing a coin four times
One coin = 2 possible out comes
4 tosses each time
Total possible combinations are 4^2 = 16
Sample space S is the set of all 16 stings of four Hâ€™s and Tâ€™s
Sample Space = sample ^ probabilities denominator
Ex:4.6
Coin - four tosses a timeDice â€“ 2 toss
2 sides6 sides
Sample = 4 tossesSample â€“ 2 tosses
Sample Space S = 2^4 = 16 Sample Space S = 6^2 = 36
-Sample space such as H/T and yes/no are categorical variables
Events: and event is an outcome or a set of outcomes of a random phenomenon. That is, an event
is a subset of the sample space.
Ex 4.8 Exactly 2 heads in four tosses.
We know total Sample space = 16, find all possibilities with 2 Hâ€™s which is 6, proportion of
getting 2 Hâ€™s is 6/16 = 0.375
Probability Rules
Rule 1: The probability P(A) of any event A satisfies 0=P(A)=1
Rule 2: If S is the sample space in a probability model, then P(S) = 1
Rule 3: Two events A and B are disjoint if they have no outcomes in common and so can never
occur together. If A and B are disjoint,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events.
Rule 4: The complement of any event A is the event that A does not occur; written as Ac. The
complement rule states that P(Ac) = 1 â€“ P(A)
Assigning probabilities: finite number of outcomes
Probability in a Finite Sample Space
Assign a probability to each individual outcome. These probabilities must be numbers between 0
and 1 and must have a sum of 1.
www.notesolution.com
The probability of any event is the sum of the probabilities of the outcomes making up the event.
Ex 4.10 Accidents on weekends
accidents on weekends â€“ P(sat or sun) = P(sat) +P(sun) = 0.02+0.03 = 0.05
Ex 4.13 How to find Benfordâ€™s law?
A = 1 P(A) = P(1) = 0.301
B = 6 p(B) = p(6) + P (7) + P(8) + P(9) = 0.222
-P(6) is included when B is = 6. But when B > 6 then P(6) is not included in the sum.
When probabilities are not disjoint, the sum addition rule cannot be applied
Ex 4.14
P( C)= p(1) + p(3) +p(5)
P(B) = p(3) + p(4) + p(5)
P( B or C) = p(1) + p(3) + p(4) + p(5)
â€¢Non disjoint probabilities are only added once
Assigning probabilities: equally likely outcomes
Equally likely outcomes
If a random phenomenon has k possible outcomes, all equally likely, then each individual
outcome has probability 1/k. The probability of any event A is
P(A) = count of outcomes in A = count of outcomes in A
count of outcomes in S k
Independence and the multiplication rule
The multiplication rule for independent events
The events A and B are not disjoint â€“ meaning the have the same outcomes
Rule 5: Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs. IF A and B are independent, P(A and B) = P(A)P(B)
Ex 4.16 Probability model that is independent is coin tossing; first toss doesnâ€™t affect the second
toss
Ex 4.17 Probability model that is dependent is picking up a card with the same color either red or
black from the deck.
Picking up a red card is 26/52 = 0.50. Once the first red card is picked, only 25 reds remain
among the 51 cards. Probability that the second card is red is 25/51 = 0.49.
â€¢Knowing the outcome of the first card deal changes the probabilities for the second.
www.notesolution.com

#### Loved by over 2.2 million students

Over 90% improved by at least one letter grade.

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah â€” University of Toronto

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim â€” University of Michigan

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna â€” University of Wisconsin

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne â€” University of California
Description
Chapter 4 4.1 Randomness Probability - a pattern that emerges clearly only after many repetitions. The proportion of times the outcome will occur in a long series of repetitions. 4.1 Toss a coin 5000 times. Probability describes only what happens in the long run. (repetitive trials) The language of probability Random in statistics is a kind of order that emerges in the long run. Randomness and probability Random when the individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions Probability (random) repetitive trials (look at overall picture) Fair has equally likely chances Ex: Fair Coin = head and tail sides Thinking about randomness - Long series of independent trials - Observing many trials (done practically) - Simulations useful when a lot trials are down without being practical The uses of probability Games of chance Ex: dice, cards, or spinning a roulette wheel, gambling Job wise Ex: astronomy, 4.2 Probability Models Probability model description of random phenomenon in the language of mathematics Description of probability outcomes (probability model): - A list of possible outcomes - A probability for each outcome Sample Space The sample Space (S) of a random phenomenon is the set of all possible outcomes. - Each possible outcome is a sample and the sample space contains all possible samples. www.notesolution.com
More Less

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Unlock Document

Unlock to view full version

Unlock Document
Notes
Practice
Earn
Me

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.