MAT 209 - Statistics Chapter 5 - Rules of Probability

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Chapter 5 Some Rules of Probability

Theory of probability/Mathematics of chance: Branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

5.1 Sample Space

Statistical experiment: Any process of observation or measurement.

Outcomes of the experiment: Results obtained from experiment (ex: readings, counts, values).

Sample space: Set of all possible outcomes.

·      Finite = counting set, limited set of possible outcomes.

·      Infinite = measuring a process, an infinite set of possible outcomes.

Attach the various possibilities to variables to make it easier to visualize the various possibilities.

5.2 Events

Event: Subset of sample space. (Usually denoted by capital letters.

·      Empty set (ø): Subset with no elements.

·      Compounded events:

§  Union – Events A and B, denoted by  event consists of all elements (outcomes) contained in event A, in event B, or in both.  = or.

§  Intersection – Events A and B, denoted by  event consists of all elements (outcomes) contained in both A and B. = and.

o   When , Events A and B are mutually exclusive, because they cannot occur at the same time.

§  Complement – Event A, denoted by A’, event that consists of all elements (outcomes) of sample space that are NOT contained in A.

Example 5.3 Number of days it rains in Chicago in January, where S = {0,1,2,3,4,…,30,31},
M = {10,11,12,…,19,20}, N = {18,19,20,…,30,31}.

a)

b)

c) M’ = {0 – 9, 21 – 31}

d) N’ = {0,1,2 … 15,16,17}

e)

f)

Another technique used to picture sample spaces and relationships among events, Venn diagram.

In Venn diagram:

- Sample space represented by rectangle.

- Events represented by regions within the rectangle (usually circles or parts of circles).

- Tinted regions represent event, its complement, union and intersection of X & Y (p.151).

- Usually used when dealing with 3 events as it is easy to determine whether or not the each of the events are contained within X, X’, Y, Y’, Z, or Z’.

5.3 Basic Rules of Probability

1.     Probabilities are real numbers between 0 and 1, inclusive.

2.     Event certain to occur, probability = 1.
Event certain NOT to occur, probability = 0.

3.     If two events are mutually exclusive, probability that one OR the other will occur equals sum of their probabilities.

4.     Sum of probabilities that an event will occur + event NOT occur = 1.

Example 5.7 If A = student will stay home to study and B = student will go to movies, P(A) = 0.64, and P(B) = 0.21 find (a) P(A’), (b)  (c) .

a) P(A’) à 1 – P(A) = 1 – 0.64 = 0.36

b)

c) Events A and B are mutually exclusive, CANNOT possibly both occur à

5.4 Probabilities and Odds

The odds and event will occur are given by the ratio of the probability that it will occur to the probability that it will NOT occur.

MATHS: If the probability of an event is p, the odds for its occurrence are a to b, where a and b are positive values such that

Example 5.9 What are the odds for occurrence of an event with probability (a) 5/9 (b) 0.85?

a) Odds are  or 5 to 4.

b) Odds are 0.85 to 0.15, 85 to 15 or if simplified 17 to 3.

*If an event is more likely NOT to occur than occur, it is customary to quote odds that it will NOT occur.

In betting, odds = ratio of wager of one party to that of another. Ex: Gambler says, “I’ll give 3 to 1 that x will happen.” = Willing to bet $3 against $1 (or higher proportions such as $3000 against $1000). IF betting odds = odds of event probability, then betting odds are FAIR.

Theoretically, if betting odds are fair, gambler should also be willing to bet on an event not occurring. Ex: Above gambler would also be willing to bet $1 against $3 (or higher proportions) that event will NOT occur.

Formula relating probability to odds: If the odds are a to b that an event will occur, the probability of its occurrence is

Subjective probabilities follow the both the 1^st and 2^nd rules of probability. However, the 3^rd rule is NOT always necessarily satisfied when it comes to subjective probabilities.

3^rd rule provides important consistency criterion to make sure subjective probabilities are valid.

Example 5.14 Economist feels that odds are 2 to 1 beef prices go up next month, 1 to 5 unchanged, and 8 to 3 that will go up or remain unchanged. Are the probabilities consistent?

Beef goes up:

Unchanged:

Goes up/remains unchanged:

 NOT consistent, judgment of the economist should be questioned.

5.5 Addition Rules

3^rd Rule of Basic Probability: (stated above) If two events are mutually exclusive, probability that one OR the other will occur equals sum of their probabilities. .
Also referred to as the “special addition rule” because of its application to ONLY mutually exclusive events à can be generalized to apply to more than 2 or non-mutually exclusive events.

Special Addition Rule for Two or More Events: If k events are mutually exclusive, the probability that one of them will occur equals the sum of their respective probabilities.

 

where A₁, A₂, … and A_k are mutually exclusive events and  is or

·      Rule for Calculating the Probability of an Event: The probability of any event A is given by the sum of the probabilities of the individual outcomes comprising A.

When all outcomes are equiprobable, leads to formula  where, n = total # of outcomes, s = number of successes.

For that NOT mutually exclusive à General Addition Rule:

* Note that when A and B are mutually exclusive that .

Example 5.19 If one card from 52 card deck drawn, what is probability that it is club OR face card?

P(C) = 13/52                        P(F) = 12/52            

So that

5.6 Conditional Probability

Conditional Probability = A measure of the probability of an event (some particular situation occurring) given that another event has occurred.

If P(B) is not equal to zero, then the conditional probability of A relative to B, namely, the probability of A given B, is .

5.7 Independent Events

In regards to conditional probability , IF event A is independent of event B. à Event A is independent of event B if the probability of event A is not affected by the occurrence or nonoccurrence of event B.

IF events A and B are NOT independent = dependent events.

5.8 Multiplication Rules

General Multiplication Rule enables us to calculate the probability that two events will BOTH occur.

ONLY when both A and B are independent can we apply the special multiplication rule:

Example 5.28 Draw one random card from deck of 52. R = event of red card, Q = event of Queen card. Are the events R and Q independent?

Find that

Sampling can be done with OR without Replacement. à Usually w/o replacement.

5.9 Bayes’s Theorem

The probability of an event, based on prior knowledge of conditions that might be related to the event. Reversing the logic of cause and effect à effect to cause. Using this logic for statistical inferences = Bayesian inferences.

Bayes’s theorem:  is the probability that event A was reached via its ith branch of the tree for (i = 1, 2, …, or k), and it can be shown that its value is given by the ratio of the probability associated with all branches.

Example 5.32 Cars are tested for their greenhouse gas emissions. 25% of all cars emit excessive greenhouse gases. When tested, 99% of all cars that emit excessive greenhouse gases will fail, BUT 17% of cars that do not emit excessive greenhouse gases also fail. What is the probability that a car that fails the test actually emits excessive greenhouse gases?

A = cars that fail the test      B = cars that emit excessive greenhouse gases (0.25)

 and

P(A) has two routes through B and B’.

P(B) = 0.25    P(A|B) = 0.99                        à (0.25)(0.99) = 0.2475

P(B’) = 0.75   P(A|B’) = 0.17                       à (0.75)(0.17) = 0.1275

Both these situations are mutually exclusive and therefore apply the 3^rd basic rule of probability.

P(A) = 0.2475 + 0.1275 = 0.3750

Using the values listed above:

P(A|B) = Probability that the car that does NOT pass emissions test, fails

P(B|A) = Probability that the car fails, does NOT pass emissions test.

Example 5.85 About 20% of residents of an equatorial city are suffering from a tropical disease. Among afflicted individuals, a medical diagnostic test accurately discloses the presence of disease 95% of the time, BUT for people, w/o disease test falsely indicates positive 10% of time. What is the probability that a person who fails the test will actually have the disease?

A = Event that person tests positive

B = Event that a person has a tropical disease (0.2)

B’ = Person does NOT have a tropical disease (0.8)

P(A|B) = Probability that person with tropical disease will test positive (0.95)

P(A|B’) = Probability that person without tropical disease will test positive (0.10)

P(A) branches:

P(B) = 0.2	P(A|B) = 0.95	P(B)*P(A|B) = (0.2)(0.95)	0.19
P(B’) = 0.8	P(A|B’) = 0.10	P(B’)*P(A|B’) = (0.8)(0.10)	0.08

P(A) = 0.19 + 0.08 = 0.27

0.704 Probability that person who tests positive has tropical disease. NOT a convincing test.