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Chapter 4

Chapter 4

10 Pages
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Department
Statistics
Course Code
STAB22H3
Professor
Moras

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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
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 Hs and Ts
Sample Space = sample ^ probabilities denominator
Ex:4.6
Coin - four tosses a timeDice – 2 toss
2 sides6 sides
Sample = 4 tossesSample – 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 Hs which is 6, proportion of
getting 2 Hs 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 – 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 – P(sat or sun) = P(sat) +P(sun) = 0.02+0.03 = 0.05
Ex 4.13 How to find Benford’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)
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 – 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 doesnt 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.
Knowing the outcome of the first card deal changes the probabilities for the second.
www.notesolution.com

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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
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