Probability

Continuous Random Variable

Is a variable that takes values within an interval.

The complement of "at least one" is ______.

"none."

|

Given

"At least one is equivalent to _________.

"one or more."

Complementary

The chance of something equals 100% minus the chance of the opposite thing.

Probability rule 2:

The more a procedure is repeated. The closer observed probability ill get to classical probability.

Union

U is called Union and is analogous to the OR event

Baye's theorem

is a way of finding a probability when we know certain other probabilities.

conditional chance

the problem puts conditions on the first card

Events that are _________ cannot occur at the same time.

disjoint

Probabilist

is a mathematician who specializes in computing the probabilities of complex events

"P"

is short for probability

Unconditional chance

the problem puts no conditions on the first card

Inclusion-Exclusion Principle

the total probability of a set of events is the sum of the individual probabilities of each individual event, minus the overlap (the probability of the events happening at the same time). In Combinatorics, it's the same principle, only instead of probabilities it is defined in terms of the union of two sets: |A ∪ B| = |A| + |B| - |A ∩ B|

P("At least 1")= 1-P(none)

1 or more so 1-("none") complement of at least 1 is 0

History

the credit of Probability's theoretical and mathematical foundations can be given to scientists like DeMoivre, Kolgomorov, Gauss, Laplace,Bernoulli, Pascal,Fermatand many more.

Simple event

An event that has only one outcome for an experiment.

Complementary events

Are two outcomes of an event that are the only two possible outcomes (mutually exclusive). This is like flipping a coin and getting heads or tails. ... Rolling a die and getting a 1 or 2 are not complementary since there are other outcomes that may happen (3, 4, 5, or 6). They cannot happen at the same time denoted with ~E/ ~A

DEPENDENT EVENT

Two events are dependent if the occurence of one is dependent on the occurrence of the other.• two or more events in which the outcome of one event affects the outcome of the other events Example :The investment choice of John's client is dependent on John's recommendation.

COMPOUND EVENT

A collection of more than one outcomes for an experiment.

Event

A collection of outcomes of a procedure

CLASSICAL/Theoretical(Axiomatic)

Is based on the counting principle. This approach assumes that every outcome is equally likely. Example lottery games, rolling a dice, coin. Probability of getting a 1 on a roll of a dice. P(A)=1/6

Probability rule 1:

Probability are always between 0 and 1 p=0 impossible event p=1 certain eventComplie

Summary -->

1) The three types of probability are classical, empirical, and subjective. 2) Classical probability uses sample spaces. 3) Empirical probability uses frequency distributions and is based on observations 4) In subjective probability, the researcher makes an educated guess about the chance of an event occurring.

Non Mutually exclusive /disjoint events: Inclusion Exclusion Principle

(A or B) =P(A) + P(B) -P(A and B) is called Intersection is analogous to the AND event Example:What is the probability of drawing a king or a spade card from the pack of cards .There are total 52 cards. There are 4 kings and 13 spade cards. One card is both a spade and a king.P (King or Spade) = 4/52 +13/52 -1/52 =16/52 Subtracting the double counting

Multiplication rule

Independent events: P(A and B) =P(A) * P(B) Example: What is the probability of drawing a king and 6 on a toss of diceThere are total 52 cards. There are 4 kings P (King and 6 from a dice) = 4/52 * 1/6 Dependent events: P(A and B) =P(A) * P(B|A) Probability of event A and B = Probability of A* Probability of B given A has occurred

OUTCOMES

Observations obtained when an experiment is performed once.

Example: What is the probability that you choose a king of diamonds given the card you choose was red.

P (Diamond_King | Red) = P(Diamond_King_and_Red)/P(Red)= 1/26/26/52= 1/13

The _________ distribution is a discrete probability distribution that applies to the number of occurrences of some event over specified interval.

possion

Three rules of Additional probability

1) To find the chance that at least one of two things will happen, check to see if they are mutually exclusive. If they are, add the chances. 2) Mutually exclusive: P(A or B) =P(A) + P(B) Probability of event A OR B P(A U B) =P(A) + P(B) U is called Union andis analogous to the OR event 3) Non-mutually exclusive: P( A or B)= P(A) + P (B) - P(A and B) ∩ is called Intersection is analogous to the AND event

Three of a Kind [aaabc]

13C1 x 4C3 x 12C2 x 4^2 Prob(Three of a kind) = ------------------------- = .02112845 52C5

Two pairs [aabbc]

13C2 x 4C2 x 4C2 x 11 x 4 Prob(2 pairs) = ------------------------- = .0475390 52C5

binomial formula

Binomial is applicable when selecting from a finite population with replacement. Rules: 1) the value of n must be fixed in advance 2) p must be the same from trial to trial 3) the trails must be independent 4) Each observation represents one of two outcomes ("success" or "failure").

Which of the following is NOT a principle of probability? A. The probability of any event is between 0 and 1 inclusive. B. The probability of an event that is certain to occur is 1. C. All events are equally likely in any probability procedure. D. The probability of nay impossible event is 0.

C. All events are equally likely in any probability procedure.

SD Calculation as a Shortcut

Calculating SD is an arduous task but it has a shortcut if there only two numbers in the list though repeated many times. (Big number - small number) square root fraction of a small number * faction of big number Example: 5,1,1,1 (5-1) square root (1/4) *(3/4) = 1.73

low probability

Rare/unusual occurrence: Pr(A)<0.05; less than 0.05%

Multiplication Rule

The chance that two things will both happen equals the chance that the first will happen, multiply by the chance that the second will happen given the first has happened

EXPERIMENT

The process of performing a task that results in one of many results.

Important to know -->

The sum of draws is likely to be around _______, give or take ___________ or so. The expected value for the sum fills in the first blank. The SE for the sum fills in the second blank.

What is the probability of drawing an ace from a shuf- fled pack of cards?

There are 4 aces. There are 52 cards in total. Therefore the probability is P( ace ) = 4/52 = 1/13

Combinations

This is different from Permutation only because for this arrangement order does not matters. The over counting in permutation has to be factored out.

A picture of line segments branching out from one starting point illustrating the possible outcomes of a procedure is called a _________.

Tree diagram

As a procedure is repeated again and gain, the relative frequency of an event tends to approach the actual probability. This is known as ______.

the law of large numbers

Number pf tosses: 10,000 heads: 5067 difference: 67

As the number of tosses are increasing the half number of tosses-number of heads is increasing but the size of the difference (67) is very small as compared to the number of tosses(10,000).To put in other words the proportion of heads -50% approaches zero.The chance error (67) is large in absolute terms but is small as a proportion of the number of trials(67/10,000).

What is randomness?

Random is part of our everyday language. We say something is random when it is unpredictable, unexpected, or out of the ordinary. Mathematics provides a framework for understanding some of these unpredictable events called probability theory.

Not mutually exclusive

can happen at the same time P(A or B)= P(A)+P(B)-P(A and B)

of

means times

The classical approach to probability requires that the outcomes are

equally likely

The __________ of a discrete random variable represents the mean value of the outcomes.

expected value

Probability can be expressed as

fraction, a decimal, or a percent

hypergeometric distribution

is a statistical experiment that has the following properties: A sample of size n is randomly selected without replacement from a population of N items. In the population, k items can be classified as successes, and N - k items can be classified as failures.

unconditional chance

the problem puts no conditions on the first cardf

Experiment

the process of measuring or observing an activity for the purpose of collecting data. ex: rolling a single 6 sided die

law of large numbers

when an experiment is conducted a large # of times, the empirical probabilities of the process will converge to the classical probabilities ex: flip a coin many times

Permutation

Is a generalization of the counting principle. This mathematically enumerates the number outcomes when r objects are selected out of n. This is an arrangement where order matters.

Probability

Is a mathematical measure of the likelihood that an event occurred. Total probability of all outcomes of an event equals 1 or 100%.

Additional rule

P(A or B) =P(A) + P(B) -P(A and B) (not mutually exclusive) P(A or B) =P(A) + P(B) (mutually exclusive) U is called Union and is analogous to the OR event

What is the probability of one pair in a 5 card hand from a well shuffled deck?

P(One Pair)= P(53 one of the 13 kinds of pairs in a deck)*P(Picking 2 suits)*P(Picking the other 3 cards from The rest of the 12 pairs)*(Pick one card from the four suits) P(one pair in a 5 card hand from a well shuffled deck) = (13 chose 1)*(4 chose 2) (12 chose 3) (4 chose 1) (4 chose 1) (4 chose 1) / (52 chose 5)

What is the probability that the second card is an Ace given that the first card is also an ace?

P(The second card is an Ace | the first card is an Ace) = (4 chose 2) / 52 chose 4) = 6/1327*100%= .5%

Flip coin 20 tines. Probability of flipping at least one head?

P*("At least 1 head")= 1-P*(no heads) = 1-P*(T and T and....T20) = 1-(1/2*1/2...1/2n20) =1-(1/1048576) =1,048,575/1048576 = .99999046

Discrete Random Variable

Is a variable that has countable possible

When Not to Add Probabilities

When the events are not mutually exclusive. P(A or B) = P(A) + P(B) − P(A and B)

Three rules of Multiplication rules: Independent events

1) Independent events: P(A and B) =P(A) * P(B) Example: What is the probability of drawing a king and 6 on a toss of diceThere are total 52 cards. There are 4 kings P (King and 6 from a dice) = 4/52 * 1/6 2) Dependent events: P(A and B) =P(A) * P(B|A) Probability of event A and B = Probability of A* Probability of B given A has occurred

One pair [aabcd]

13 x 4C2 x 12C3 x 4^3 Prob(1 pair) = ---------------------- = 0.422569 52C5

Complimentary events are necessarily mutually exclusive events but vise versa is not necessarily true. Why?

Because complementary events are two outcomes of an event that are the only two possible outcomes, whereas mutually exclusive events can be multiple events.

mutually exclusive

Events that cannot occur at the same time. The occurrence of one prevents the occurrence of the other: one excludes the other. Ex: cards, penny can't overlap can't happen at the same time P(A & B)=0

SUBJECTIVE

The one time probability estimation of an event by the personal degree of belief of the happening of an event. Example probability of a stock going up the next day. The experiment cannot be repeated therefore the relative frequency approach cannot be used. This is the subjectivist approach.

Factorial

Where Factorial r (r!) can be denoted as follows: r! = (r)(r-1)(r-2)...................(3)(2)(1)

Law of average

as the number of trials increases, the empirical probability of will approach the theoretical probability

INDEPENDENT EVENT

•Two events are independent if the occurence of one is not dependent on the occurrence of the other the occurrence of one event does not effect the occurrence of the others e.g if we flip a coin two times, the first time may show a head, but the next time when we flip the coin the outcome will be heads also. From this example we can see the first event does not affect the occurrence of the next event. •Example:The investment incurred by a company is independent of the investment incurred by another company

Three Rules of Probability

1) For any sample space Ω={A1,A2,................An}1) 2) Probability of each event is between 0 and 3) For two disjoint events A and B the probability of either of the event occurring is P(A or B)=P(A)+P(A)

what is a tree diagram?

A device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment

hypergeometric formula

A hypergeometric experiment has two distinguishing characteristics: The researcher randomly selects, without replacement, a subset of items from a finite population. Each item in the population can be classified as a success or a failure.

Can I get an example?

Dice, cards, coin flips, spinners are common examples of randomizing tools we use in board games and gambling. We use them because they are unpredictable. A fair, six-sided dice will generate a number from one to six with an equal probability of each number. Similarly, we consider coin flip to be fair if there is an equal probability of arriving at heads or tails.

Someone throws a pair of dice. True or false, the chance of getting at least one ace is 1/6 + 1/6 = 1/3

False, imagine one of the dice is white, and other is black. A white ace does not prevent a black ace. These two events are not mutually exclusive, so the additional rule does not apply. Adding gives the wrong awnser.

TREE RULES

Helps implement conditional probabilities it partitions the space into disjoint events.

Additional rules: Mutually exclusive /disjoint events

If S or Ωis a Sample Space and A AND B are two events thenP(A or B) =P(A) + P(B) Probability of event A OR B P(A U B) =P(A) + P(B)U is called Union and is analogous to the OR event

Complimentary Event

If only two events encompass up the whole sample space. The two events are complimentary to each other. P(E)+P(~E) =1 Or P(E)+P(Ec) =1

Fundamental Principle of Counting

If there are n1 possible outcomes of the first experiment , n2possible outcomes of the second experiment then the total outcomes of the composite experiment are n1 times n2 Example: If John chooses from amongst the four companies and then differentiates his choice as per the four divisions then in total how many choices does he have? Answer: Using the counting principle he has 4 (companies) times 4 (divisions) =16 possible choices. security code * first number cant be zero 9*10*10*10= 9000 combinations

Single event

Is a unique possible outcome of a random circumstance. For an experiment, an event to be any collection of possible outcomes. Any particular outcome is known as a simple event.

RELATIVE FREQUENCY/EMPIRICAL

Is based on performing the experiment multiple times and calculating the relative frequencies which depicts probability. Long run relative frequency is the probability of an event. Example Checking a production process, tossing a coin. This is the Frequentist's approach. Example: If we roll a unbiased coin 100 times and if we get 55 heads then the relative frequency = number of heads/total trials = 55/100. If this experiment is done for a large number of times then this relative frequency approached the probability of the event.

CHANCE ERROR

Is the difference in the sample statistics over many draws with replacement due to sample to sample variability in random processes. Example: Looking at the total tosses minus the number of heads:

Probability Distribution

Is the mathematical function that describes the possible values of the random variable and their associated probabilities.

RANDOM VARIABLE

Is the numerical representation of an event in the sample space. Its value varies due to chance occurrence of the outcome.It is generically denoted by the alphabetical letter X or Y X could be1,2,3,4,5,6.

Adding Probabilities

It provided A and B cannot happen together. A and B must be mutually exclusive outcomes. What is the probability of drawing an ace or a king from a shuffled pack of cards? P( ace ) = 1/13 P( king ) = 1/13 ⇒ P( ace or king ) = 1/13 + 1/13 = 2/13

Find the probability that if a couple has 3 kids, two will be boys. (assuming equal chance of boy/girl)

Procedure: Having 3 children Event: 2 boys/ 1 girl Sample size {BBB, BBG, BGB, BGG, GGG, GGB, GBG, GBB} Math: .5^2 * .5^1 * 3 = 3/8 = 0.375

What is the probability of a royal flush hand from a well shuffled deck?

Royal Flush is {10, J, Q, K, A}of the same suitWe need to choose one suit(same color) 1 x 4C1 Prob(Royal flush) = ---------- = .000001539 52C5

SAMPLE SPACE

SPACEThe set of all the results that are obtained when an experiment is performed.Tossing a dice Sample Space Ω: {1 ,2, 3, 4, 5, 6} Is denoted by the symbol omega Ω

We expect what, exactly?

Since we have assigned values to different events, $1 for heads and $-1 for tails, we can ask "What is average value?" That is, if we repeated this game for a long time, what would be the average change of money be per game? For any finite number of coin flips, this average is called the sample average. There is a theorem called the law of large numbers. The gist of it is that as the number of flips increases, the sample average will get closer and closer to something called the expected value. For a coin flip, we calculate the expected value by (1)(p) + (-1)(1-p). To calculate expected value for discrete events, like dice rolls and coin flips, we multiply the value by the probability and add up all possibilities. \sum_S v_s P[S] So, for our example, we have (1)(.5) + (-1)(.5), which equals 0. That means, if we do a long enough experiment, the average value each coin flip should be about 0.

What is the difference between hypergeometric and binomial?

The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of k successes in n draws with replacement.

Conditional probability

The probability of event B occurring given that event A has already occurred/is occurring. This is the dependent event case but can be written mathematically using the following equation:

LAW OF LARGE NUMBERS/(In formally LAW OF AVERAGES)

The sample mean/proportion approaches closer and closer to the population mean/proportion as the sample size approaches the population size. Law of averages does not apply to the count of the sample statistics .It is not true that Tossing a fair coin a large number of times will result in exactly or about half heads and half tails. It is the proportion of heads that in the long run will be ½.The law of averages should be thought of in terms of limits as n tends to infinity rather than in terms of finite sample sized simulation. LtHnhttp://www.timswast.com/blog/2013/07/28/animated-demonstration-of-probability/ Example: Tossing a coin a large number of times.

MUTUALLY EXCLUSIVE EVENTS/DISJOINT

Two events are mutually exclusive if the happening of one precludes the happening of the other. two events are mutually exclusive event when they cannot occur at the same time. e.g if we flip a coin it can only show a head OR a tail, not both. Example: The investment incurred by a company is independent of the investment incurred by another company

Baye's theorem with multiple variables

https://www.youtube.com/watch?v=P6n_Yf7EUX0