Arithmetic Reasoning and Mathematics Knowledge Review: Probability and Data Analysis

Probability is the likelihood of a certain outcome occurring for a given event. An event is a situation that produces a result; that 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 probability can usually be determined without actually performing the event. The likelihood of a 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%).

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

, 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

. So

. By similar reasoning, if A and B are independent then

.

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 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.


P1. Determine the theoretical probability of the following events:

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.
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

.

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

.

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:
 

Roll - Cards - Roll - Probability of Card - Probability of Event
1 -

 


2
1




3
1,2




1,2,3




1,2,3,4




6
1,2,3,4,5




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:

.

Data Analysis

Mean
The statistical mean of a group of data is the same as the arithmetic average of that group. To find the mean of a set of data, first convert each value to the same units, if necessary. Then find the sum of all the values, and count the total number of data values, making sure you take into consideration each individual value. If a value appears more than once, count it more than once. Divide the sum of the values by the total number of values and apply the units, if any. Note that the mean does not have to be one of the data values in the set, and may not divide evenly.


For instance, the mean of the data set {88, 72, 61, 90, 97, 68, 88, 79, 86, 93, 97, 71, 80, 84, 89} would be the sum of the fifteen numbers divided by 15:





While the mean is relatively easy to calculate and averages are understood by most people, the mean can be very misleading if used as the sole measure of central tendency. If the data set has outliers (data values that are unusually high or unusually low compared to the rest of the data values), the mean can be very distorted, especially if the data set has a small number of values. If unusually high values are countered with unusually low values, the mean is not affected as much. For example, if five of twenty students in a class get a 100 on a test, but the other 15 students have an average of 60 on the same test, the class average would appear as 70. Whenever the mean is skewed by outliers, it is always a good idea to include the median as an alternate measure of central tendency.

A weighted mean, or weighted average, is a mean that uses 'weighted' values. The formula is

. Weighted values, such as

 are assigned to each member of the set

. If calculating weighted mean, make sure a weight value for each member of the set is used.

Median
The statistical median is the value in the middle of the set of data. To find the median, list all data values in order from smallest to largest or from largest to smallest. Any value that is repeated in the set must be listed the number of times it appears. If there are an odd number of data values, the median is the value in the middle of the list. If there is an even number of data values, the median is the arithmetic mean of the two middle values.
For example, the median of the data set {88, 72, 61, 90, 97, 68, 88, 79, 86, 93, 97, 71, 80, 84, 88} is 86 since the ordered set is {61, 68, 71, 72, 79, 80, 84, 86, 88, 88, 88, 90, 93, 97, 97}.
The big disadvantage of using the median as a measure of central tendency is that is relies solely on a value's relative size as compared to the other values in the set. When the individual values in a set of data are evenly dispersed, the median can be an accurate tool. However, if there is a group of rather large values or a group of rather small values that are not offset by a different group of values, the information that can be inferred from the median may not be accurate because the distribution of values is skewed.

Mode
The statistical mode is the data value that occurs the greatest number of times in the data set. It is possible to have exactly one mode, more than one mode, or no mode. To find the mode of a set of data, arrange the data like you do to find the median (all values in order, listing all multiples of data values). Count the number of times each value appears in the data set. If all values appear an equal number of times, there is no mode. If one value appears more than any other value, that value is the mode. If two or more values appear the same number of times, but there are other values that appear fewer times and no values that appear more times, all of those values are the modes.
For example, the mode of the data set {88, 72, 61, 90, 97, 68, 88, 79, 86, 93, 97, 71, 80, 84, 88} is 88.
The main disadvantage of the mode is that the values of the other data in the set have no bearing on the mode. The mode may be the largest value, the smallest value, or a value anywhere in between in the set. The mode only tells which value or values, if any, occurred the greatest number of times. It does not give any suggestions about the remaining values in the set.

Range
The range of a set of data is the difference between the greatest and lowest values of the data in the set. To calculate the range, you must first make sure the units for all data values are the same, and then identify the greatest and lowest values. If there are multiple data values that are equal for the highest or lowest, just use one of the values in the formula.
Write the answer with the same units as the data values you used to do the calculations.