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Mathematics: Probability

Probability
Probability is the likelihood of a certain outcome occurring for a given event. An event is any situation that produces a result. It could be something as simple as flipping a coin or as complex as launching a rocket. Determining the probability of an outcome for an event can be equally simple or complex. As such, there are specific terms used in the study of probability that need to be understood:

Compound event – event that involves two or more independent events (rolling a pair of dice and taking the sum)
Desired outcome (or success) – an outcome that meets a particular set of criteria (a roll of 1 or 2 if we are looking for numbers less than 3)
Independent events – two or more events whose outcomes do not affect one another (two coins tossed at the same time)
Dependent events – two or more events whose outcomes affect one another (two cards drawn consecutively from the same deck)
Certain outcome – probability of outcome is 100% or 1
Impossible outcome – probability of outcome is 0% or 0
Mutually exclusive outcomes – two or more outcomes whose criteria cannot all be satisfied in a single event (a coin coming up heads and tails on the same toss)
Random variable – refers to all possible outcomes of a single event which may be discrete or continuous.

Theoretical and Experimental Probability
Theoretical probability can usually be determined without actually performing the event. The likelihood of an outcome occurring, or the probability of an outcome occurring, is given by the formula:

Note that P(A) is the probability of an outcome A occurring, and each outcome is just as likely to occur as any other outcome. If each outcome has the same probability of occurring as every other possible outcome, the outcomes are said to be equally likely to occur.

The total number of acceptable outcomes must be less than or equal to the total number of possible outcomes. If the two are equal, then the outcome is certain to occur and the probability is 1. If the number of acceptable outcomes is zero, then the outcome is impossible and the probability is 0. For example, if there are 20 marbles in a bag and 5 are red, then the theoretical probability of randomly selecting a red marble is 5 out of 20,

, 0.25, or 25%).

If the theoretical probability is unknown or too complicated to calculate, it can be estimated by an experimental probability. Experimental probability, also called empirical probability, is an estimate of the likelihood of a certain outcome based on repeated experiments or collected data. In other words, while theoretical probability is based on what should happen, experimental probability is based on what has happened.

Experimental probability is calculated in the same way as theoretical, except that actual outcomes are used instead of possible outcomes. The more experiments performed or datapoints gathered, the better the estimate should be.
Theoretical and experimental probability do not always line up with one another. Theoretical probability says that out of 20 coin-tosses, 10 should be heads. However, if we were actually to toss 20 coins, we might record just 5 heads. This doesn't mean that our theoretical probability is incorrect; it just means that this particular experiment had results that were different from what was predicted. A practical application of empirical probability is the insurance industry. There are no set functions that define lifespan, health, or safety. Insurance companies look at factors from hundreds of thousands of individuals to find patterns that they then use to set the formulas for insurance premiums.

Objective and Subjective Probability
Objective probability is based on mathematical formulas and documented evidence. Examples of objective probability include raffles or lottery drawings where there is a pre-determined number of possible outcomes and a predetermined number of outcomes that correspond to an event.
Other cases of objective probability include probabilities of rolling dice, flipping coins, or drawing cards. Most gambling games are based on objective probability.
In contrast, subjective probability is based on personal or professional feelings and judgments. Often, there is a lot of guesswork following extensive research. Areas where subjective probability is applicable include sales trends and business expenses. Attractions set admission prices based on subjective probabilities of attendance based on varying admission rates in an effort to maximize their profit.

Sample Space
The total set of all possible results of a test or experiment is called a sample space, or sometimes a universal sample space. The sample space, represented by one of the variables S, Ω, or U (for universal sample space) has individual elements called outcomes. Other terms for outcome that may be used interchangeably include elementary outcome, simple event, or sample point. The number of outcomes in a given sample space could be infinite or finite, and some tests may yield multiple unique sample sets. For example, tests conducted by drawing playing cards from a standard deck would have one sample space of the card values, another sample space of the card suits, and a third sample space of suit-denomination combinations.

For most tests, the sample spaces considered will be finite.

An event, represented by the variable E, is a portion of a sample space. It may be one outcome or a group of outcomes from the same sample space. If an event occurs, then the test or experiment will generate an outcome that satisfies the requirement of that event.

For example, given a standard deck of 52 playing cards as the sample space, and defining the event as the collection of face cards, then the event will occur if the card drawn is a J, Q, or K. If any other card is drawn, the event is said to have not occurred.

For every sample space, each possible outcome has a specific likelihood, or probability, that it will occur. The probability measure, also called the distribution, is a function that assigns a real number probability, from zero to one, to each outcome. For a probability measure to be accurate, every outcome must have a real number probability measure that is greater than or equal to zero and less than or equal to one. Also, the probability measure of the sample space must equal one, and the probability measure of the union of multiple outcomes must equal the sum of the individual probability measures.

Probabilities of events are expressed as real numbers from zero to one. They give a numerical value to the chance that a particular event will occur.

The probability of an event occurring is the sum of the probabilities of the individual elements of that event.

For example, in a standard deck of 52 playing cards as the sample space and the collection of face cards as the event, the probability of drawing a specific face card is
, but the probability of drawing any one of the twelve face cards is
.

Note that rounding of numbers can generate different results. If you multiplied 12 by the fraction

before converting to a decimal, you would get the answer
.

Tree Diagram
For a simple sample space, possible outcomes may be determined by using a tree diagram or an organized chart. In either case, you can easily draw or list out the possible outcomes.

For example, to determine all the possible ways three objects can be ordered, you can draw a tree diagram:


You can also make a chart to list all the possibilities:





Either way, you can easily see there are six possible ways the three objects can be ordered.

If two events have no outcomes in common, they are said to be mutually exclusive. For example, in a standard deck of 52 playing cards, the event of all card suits is mutually exclusive to the event of all card values.

If two events have no bearing on each other so that one event occurring has no influence on the probability of another event occurring, the two events are said to be independent.

For example, rolling a standard six-sided die multiple times does not change that probability that a particular number will be rolled from one roll to the next. If the outcome of one event does affect the probability of the second event, the two events are said to be dependent.

For example, if cards are drawn from a deck, the probability of drawing an ace after an ace has been drawn is different than the probability of drawing an ace if no ace (or no other card, for that matter) has been drawn.

In probability, the odds in favor of an event are the number of times the event will occur compared to the number of times the event will not occur.

To calculate the odds in favor of an event, use the formula
) is the probability that the event will occur. Many times, odds in favor is given as a ratio in the form <br><img src=" />a:b, where a is the probability of the event occurring and b is the complement of the event, the probability of the event not occurring. If the odds in favor are given as 2:5, that means that you can expect the event to occur two times for every 5 times that it does not occur. In other words, the probability that the event will occur is

.
In probability, the odds against an event are the number of times the event will not occur compared to the number of times the event will occur. To calculate the odds against an event, use the formula
) is the probability that the event will occur. Many times, odds against is given as a ratio in the form <br><img src=" />b:a, where b is the probability the event will not occur (the complement of the event) and a is the probability the event will occur.

If the odds against an event are given as 3:1, that means that you can expect the event to not occur 3 times for every one time it does occur. In other words, 3 out of every 4 trials will fail.

Permutations and Combinations
When trying to calculate the probability of an event using the

formula, you may frequently find that there are too many outcomes to individually count them. Permutation and combination formulas offer a shortcut to counting outcomes.

 

A permutation is an arrangement of a specific number of a set of objects in a specific order.

The number of permutations of r items given a set of n items can be calculated as . Combinations are similar to permutations, except there are no restrictions regarding the order of the elements.

While ABC is considered a different permutation than BCA, ABC and BCA are considered the same combination.

The number of combinations of r items given a set of n items can be calculated as

or
.

Suppose you want to calculate how many different 5-card hands can be drawn from a deck of 52 cards. This is a combination since the order of the cards in a hand does not matter. There are 52 cards available, and 5 to be selected.

Thus, the number of different hands is
.

Union and Intersection of Two Sets of Outcomes
If A and B are each a set of elements or outcomes from an experiment, then the union (symbol
) of the two sets is the set of elements found in set A or set B. 

For example, if
(symbol <br><img data-cke-saved-src=" />) of two sets is the set of outcomes common to both sets. For the above sets A and B,
. For statistical events, the intersection is equivalent to 'and';

is the same thing as
.

It is important to note that union and intersection operations commute. That is:


Complement of an Event
Sometimes it may be easier to calculate the possibility of something not happening, or the complement of an event.

Represented by the symbol , the complement of A is the probability that event A does not happen.
When you know the probability of event A occurring, you can use the formula
, where

is the probability of event A not occurring, and

is the probability of event A occurring.

Addition Rule
The addition rule for probability is used for finding the probability of a compound event. Use the formula
, where

is the probability of both events occurring to find the probability of a compound event.

The probability of both events occurring at the same time must be subtracted to eliminate any overlap in the first two probabilities.

Conditional probability
Given two events A and B, the conditional probability

is the probability that event B will occur, given that event A has occurred. The conditional probability cannot be calculated simply from

and
; these probabilities alone do not give sufficient information to determine the conditional probability. It can, however, be determined if you are also given the probability of the intersection of events A and B,
, the probability that events A and B both occur. Specifically,
.

For instance, suppose you have a jar containing two red marbles and two blue marbles, and you draw two marbles at random.

Consider event A being the event that the first marble drawn is red, and event B being the event that the second marble drawn is blue.

is
, and

is
.  (The latter may not be obvious, but may be determined by finding the product of

and
).

Therefore
.

Conditional Probability in Everyday Situations
Conditional probability often arises in everyday situations in, for example, estimating the risk or benefit of certain activities. The conditional probability of having a heart attack given that you exercise daily may be smaller than the overall probability of having a heart attack. The conditional probability of having lung cancer given that you are a smoker is larger than the overall probability of having lung cancer. Note that changing the order of the conditional probability changes the meaning: the conditional probability of having lung cancer given that you are a smoker is a very different thing from the probability of being a smoker given that you have lung cancer. In an extreme case, suppose that a certain rare disease is caused only by eating a certain food, but even then, it is unlikely.
Then the conditional probability of having that disease given that you eat the dangerous food is nonzero but low, but the conditional probability of having eaten that food given that you have the disease is 100%!

Independence
The conditional probability

is the probability that event B will occur given that event A occurs. If the two events are independent, we do not expect that whether or not event A occurs should have any effect on whether or not event B occurs. In other words, we expect
.

This can be proven using the usual equations for conditional probability and the joint probability of independent events. The conditional probability
are independent, then <br><img data-cke-saved-src=" />. So
. By similar reasoning, if A and B are independent then
.

Two-way Frequency Tables
If we have a two-way frequency table, it is generally a straightforward matter to read off the probabilities of any two events A and B, as well as the joint probability of both events occurring,
.

We can then find the conditional probability

by calculating
.

We could also check whether or not events are independent by verifying whether
.

For example, a certain store's recent T-shirt sales:




Suppose we want to find the conditional probability that a customer buys a black shirt (event A), given that the shirt he buys is size small (event B). From the table, the probability

that a customer buys a small shirt is
. The probability

that he buys a small, black shirt is
. The conditional probability

that he buys a black shirt, given that he buys a small shirt, is therefore
.
Similarly, if we want to check whether the event a customer buys a blue shirt, A, is independent of the event that a customer buys a medium shirt, B. From the table,

and
. Also,
. Since
,

and these two events are indeed independent.

Multiplication Rule
The multiplication rule can be used to find the probability of two independent events occurring using the formula
, where

is the probability of two independent events occurring,

is the probability of the first event occurring, and

is the probability of the second event occurring.

The multiplication rule can also be used to find the probability of two dependent events occurring using the formula
, where

is the probability of two dependent events occurring and

is the probability of the second event occurring after the first event has already occurred. Before using the multiplication rule, you MUST first determine whether the two events are dependent or independent.

Use a combination of the multiplication rule and the rule of complements to find the probability that at least one outcome of the element will occur.

This is given by the general formula
. For example, to find the probability that at least one even number will show when a pair of dice is rolled, find the probability that two odd numbers will be rolled (no even numbers) and subtract from one. You can always use a tree diagram or make a chart to list the possible outcomes when the sample space is small, such as in the dice-rolling example, but in most cases it will be much faster to use the multiplication and complement formulas.

Expected Value
Expected value is a method of determining expected outcome in a random situation. It is really a sum of the weighted probabilities of the possible outcomes. Multiply the probability of an event occurring by the weight assigned to that probability
(such as the amount of money won or lost).

A practical application of the expected value is to determine whether a game of chance is really fair. If the sum of the weighted probabilities is equal to zero, the game is generally considered fair because the player has a fair chance to at least break even. If the expected value is less than zero, then players lose more than they win. For example, a lottery drawing might allow the player to choose any three-digit number, 000–999. The probability of choosing the winning number is 1:1000. If it costs $1 to play, and a winning number receives $500, the expected value is
. You can expect to lose on average 50 cents for every dollar you spend.

Expected Value and Simulators
A die roll simulator will show the results of n rolls of a die. The result of each die roll may be recorded. For example, suppose a die is rolled 100 times. All results may be recorded. The numbers of 1s, 2s, 3s, 4s, 5s, and 6s, may be counted. The experimental probability of rolling each number will equal the ratio of the frequency of the rolled number to the total number of rolls. As the number of rolls increases, or approaches infinity, the experimental probability will approach the theoretical probability of .

Thus, the expected value for the roll of a die is shown to be

, or 3.5.

 

P1. Determine the theoretical probability of the following events:
(a) Rolling an even number on a regular 6-sided die.
(b) Not getting a red ball when selecting one from a bag of 3 red balls, 4 black balls, and 2 green balls.
(c) Rolling a standard die and then selecting a card from a standard deck that is less than the value rolled.
P2. There is a game of chance involving a standard deck of cards that has been shuffled and then laid on a table. The player wins $10 if they can turn over 2 cards of matching color (black or red), $50 for 2 cards with matching value (A-K), and $100 for 2 cards with both matching color and value. What is the expected value of playing this game?

P3. Today, there were two food options for lunch at a local college cafeteria. Given the following survey data, what is the probability that a junior selected at random from the sample had a sandwich?




P1. (a). The values on the faces of a regular die are 1, 2, 3, 4, 5, and 6. Since three of these are even numbers (2, 4, 6), The probability of rolling an even number is
.

(b) The bag contains a total of 9 balls, 6 of which are not red, so the probability of selecting one non-red ball would be
.

(c) In this scenario, we need to determine how many cards could satisfy the condition for each possible value of the die roll. If a one is rolled, there is no way to achieve the desired outcome, since no cards in a standard deck are less than 1. If a two is rolled, then any of the four aces would achieve the desired result. If a three is rolled, then either an ace or a two would satisfy the condition, and so on.
Note that any value on the die is equally likely to occur, meaning that the probability of each roll is .

 

Putting all this in a table can help:




Assuming that each value of the die is equally likely, then the probability is the sum of the probabilities of each way to achieve the desired outcome:
.

P2. First, determine the probability of each way of winning. In each case, the fist card simply determines which of the remaining
51 cards in the deck correspond to a win. For the color of the cards to match, there are 25 cards remaining in the deck that match the color of the first, but one of the 25 also matches the value, so only 24 are left in this category. For the value of the cards to match, there are 3 cards remaining in the deck that match the value of the first, but one of the three also matches the color, so only 2 are left in this category. There is only one card in the deck that will match both the color and value. Finally, there are 24 cards left that don't match at all.

Now we can find the expected value of playing the game, where we multiply the value of each event by the probability it will occur and sum over all of them:

This game therefore has an expected value of $8.63 each time you play, which means if the cost to play is less than $8.63 then you would, on average, gain money. However, if the cost to play is more than $8.63, then you would, on average, lose money.