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

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 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.

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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.

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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.

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