Statistics Chapter 3
Addition Rule of Probability
a rule for finding the union of one event or another event happening Cheyenne and her friends are playing a board game. Each person roles a die on his or her turn and moves the number of spaces indicated on the die. Some of the spaces ask the player to pull a card from a deck of regular playing cards. Cheyenne needs to move three or six spaces to get to the next card space. Then, she needs to pick a black card or a seven. What is the probability of Cheyenne rolling a 3 or a 6 on the die? What is the probability of Cheyenne picking a card that is a black card or a seven?
Random variable
a variable that is subject to randomness, which means it can take on different values Let's see an example. We'll start with tossing coins. I want to know how many heads I might get if I toss two coins. Since I only toss two coins, the number of heads I could get is zero, one, or two heads. So, I define X (my random variable) to be the number of heads that I could get. In this case, each specific value of the random variable - X = 0, X = 1 and X = 2 - has a probability associated with it. When the variable represents isolated points on the number line, such as the one below with 0, 1 or 2, we call it a discrete random variable.
Variable
an alphabetical character that represents an unknown number Ex: If you have ever taken an algebra class, you probably learned about different variables like x, y and maybe even z. Some examples of variables include x = number of heads or y = number of cell phones or z = running time of movies.
combination
an arrangement of objects where order does not matter Ex: The Jackson Wildcats play basketball in a highly competitive city district. There are eight teams in the district, and they all play each other once during the season. The coach of the Wildcats wants to know how many games will be played in the district this season. To calculate this amount, he will need to use a combination. The coach knows that there are eight teams, but the order the teams play each other does not matter.
bivariate data
deals with two variables that can change and are analyzed together to find relationships between them For example, if Mindy was studying for a college test and tracks her study time and her test scores, she might see that the more time she spends studying, the better her test scores become. Therefore, in this scenario, Mindy's test scores are the dependent variable because they depend on the number of hours she studies. Likewise, the number of study hours would be considered the independent variable. For that reason, we can see the relationship in this bivariate data set:
dependent events
events in which the previous attempts affect the outcome of subsequent events. An example of a dependent event would be selecting a card from a deck of cards and not replacing the card. Then you draw another card from the now-smaller deck of cards. There are a lot of scenarios that could happen here. Big Bertha could get first place and another horse could get second place. Similarly, Sleepy Sally could get second place and another horse could get first place. Therefore, we could say that the probability of Sleepy Sally getting second place is dependent upon Big Bertha getting first place. Then these events would be dependent and not independent events.
mutually exclusive events
events that cannot happen at the same time Ex: This Venn diagram represents the mutually exclusive events of rolling a 3 or a 6. Notice that there is no overlap between the two circles because the events cannot happen at the same time.
continuous random variable
random variables that are found from measuring - like the height of a group of people or distance traveled while grocery shopping or student test scores. Let's look at a hypothetical table of the random variable X and the number of people who scored in those different intervals: Test ScoresFrequency(% of students)0 to <20%520% to <40%2040% to <60%3060% to <80%3580% to 100%10
Relative frequency
ratio of the occurrence of a singular event and the total number of outcomes https://study.com/academy/lesson/relative-frequency-classical-approaches-to-probability-lesson-quiz.html
Joint relative frequency
the ratio of the frequency in a particular category and the total number of data values Represented by purple cells on right table
Marginal relative frequency
the ratio of the sum of the joint relative frequency in a row or column and the total number of data values Represented by green cells on right table
Negative correlation
where the dependent variables and independent variables in a data set either increase or decrease opposite from one another
Positive correlation
where the dependent variables and independent variables in a data set increase or decrease together. Notice that most of the points increase both vertically and horizontally. You may notice that we have graphed the number of reading hours on the x-axis, horizontally, and the test scores on the y-axis, vertically. When a bivariate data set shows an overall increase in numbers like this, it is called a positive correlation, where the dependent variables and independent variables in a data set increase or decrease together.
Non-mutually exclusive events
which are events that can happen separately or at the same time Picking a card out of a deck that is black or a seven is an example of non-mutually exclusive events, which are events that can happen separately or at the same time. Notice that there is a place where the two circles overlap. This is because it is possible to pick a black card that is a seven. This is an example of non-mutually exclusive events because the two events can occur at the same time. Some of the spaces ask the player to pull a card from a deck of regular playing cards. Cheyenne needs to move three or six spaces to get to the next card space. Then, she needs to pick a black card or a seven.
Two-way table
A two-way or contingency table is a statistical table that shows the observed number or frequency for two variables, the rows indicating one category and the columns indicating the other category. The row category in this example is gender - male or female. The column category is their color.
Properties of a Binomial Experiment
First, the outcomes must be independent. This means that the outcome of one trial cannot have any influence on another. Second, a binomial experiment must only have two possible outcomes (Success or failure) Third, there are a fixed number of trials in a binomial experiment. In Dakota's experiment, there are ten people he will call - this is a fixed number that he and his friends have determined before the experiment begins.
Binomial probability formula
First, we need to find the values of x, n, and P. The x represents the number of successes, the n represents the number of trials, and the P represents the probability of success on an individual trial. We can use this information to find the probability of certain numbers of success for an experiment
bell hooks on oppression
From the perspective of hooks, we all live with the legacy of a system that has historically been dominated by those who are privileged, such as those who are rich, who are white and who are male. These systems of oppression affect all that we experience, from how each of us view ourselves to how we treat others, she says.
Binomial distribution table
Given the number of trials in an experiment ''n'', you can use these binomial distribution tables to look up the probability of a certain number of successes in the whole experiment. Each row represents ''x'', the number of successes in the total experiment, and each column represents ''P'', the probability of an individual trial's success.
Acknowledge the impact Wollstonecraft's essay had on feminism and the controversy associated with it after her death
The book's initial reviews in magazines like The General Magazine, The Literary Magazine and New York Magazine were positive, and it was published in America as well as translated into French. However, there was also negative backlash associated with the work, especially after Wollstonecraft's death in 1797. Her husband, William Godwin, published a memoir of her life in 1798, in which he revealed previously unknown facts about her, including her love affairs, her illegitimate child and her attempts at suicide. Though holding many of the same beliefs, female writers, like novelist Maria Edgeworth, hesitated to mention or reference her in their own work because her philosophies were becoming associated with her scandalous lifestyle.
Multiplication Rule of probability
Multiplying the probabilities of two events to find the total probability of them both occurring instead of one or the other
Do theoretical probability and actual probability problems
Note: If you have the probability of a certain outcome without actually doing the experiment first, then you are working with theoretical probability. If you are working with numbers from a data set based on an experiment, then you are working with actual probability.
addition rule for mutually exclusive events
P(A or B) = P(A) + P(B) Probability of Event A or B occurring=Probability of Event A occurring + Probability of Event B occurring
Conditional probability formula
P(B|A) = P(A and B) / P(A) P(B|A)=probability of B occurring given A, meaning what's the probability of B if event A also happens
bell hooks on pop culture
She also cites pop culture and critiques it and encourages her students to do so. Why? Media images are a massive part of what influences our view of the world, and they affect how we deal with social problems. This makes it all the more important to critique, hooks would argue.
Union
To collect sets together, we use the term union. We unite the sets into one. Let's say I have two sets. Set A is green, blue, and pink. Set B is orange, yellow, and black. A u B represents the union of sets A and B. Yes, that u symbol represents union! It's kind of handy! A u B represents all the elements that are listed in set A, or in set B, or in both. How would that look in using mathematical symbols? A u B = {green, blue, pink, orange, yellow, black}.
Intersection
To find elements in common with sets, we use the term intersection. Think of the sets as two roads that meet at an intersection. What do the two roads, or sets, have in common? Let's say I have two sets. Set A is 4, 6, and 9. Set B is 7, 8, and 9. A intersect B represents the intersection of sets A and B. Yes, that upside down u represents intersection! This represents all the elements that are the same in A and B. How would that look in using mathematical symbols? A intersect B = {9}.
Independent probability*
When Andrew grabs a tie out of his closet without looking, this is an example of independent probability. In this case, you only have one event to consider. Independent probability is when the probability of an event is not affected by a previous event.
independent events
When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. Meanwhile, William's friend Derek is watching his favorite basketball team play in the final game of their regional tournament. William promises Derek they will have a big celebration if Big Bertha wins first and Derek's team wins the tournament. Derek's basketball team has no relation or influence on Big Bertha's performance, and because of this, these are two independent events.
Sample space, Event, and subset
When you conduct an experiment, you are observing certain outcomes. For example, you may be conducting an experiment on flipping a coin. The possible outcomes for flipping a coin are heads or tails. If you were rolling a six-sided die, then the possible outcomes would be 1, 2, 3, 4, 5 or 6. There are no other possible outcomes. We call these possible outcomes sample spaces. A sample space is a set (S) of an experiment that includes all possible outcomes of the experiment. An event is a possible outcome of an experiment. And a subset is an event of a sample space.
False sensibility
Wollstonecraft warned against false sensibility, a tendency of women to become too overtaken by emotional sensitivity.
frequency table
a chart that shows the popularity or mode of a certain type of data Ex: The data set for the steak tasting is as follows, where each number represents the steak that was chosen as the best: 1, 5, 3, 1, 2, 3, 4, 5, 1, 4, 2, 4, 4, 5, 1, 4, 2, 4, 2, 2 We can use our data to create a frequency table like this.
Set
a collection of objects C = {pants, t-shirt, skirt, and dress} R = {...-3, -2, -1, 0, 1, 2, 3...}
Density curve
a plot of the relative frequencies of a continuous random variable
Event
a possible outcome of an experiment
discrete random variable
a random variable that represents numbers found by counting. For example: number of marbles in a jar, number of students present or number of heads when tossing two coins. Suppose I'm looking at the number of defective tires on the car. Let X = the number of defective tires on the car. Is X discrete or continuous? Well, since there are usually four tires on the car, X can range from 0-4. However, it can only be 0, 1, 2, 3 or 4. So X is a discrete random variable.
Abby is attending her first swimming competition. There are seven girls racing in the first heat. She has to place first or second to make it to the next level of the tournament. Assuming there are no ties, what is the probability Abby will get first or second?
Abby has a 2 in 7 or approximately 29% chance of making it into the next level of the tournament. This is another example of mutually exclusive events. Abby can't get both first and second place. Therefore, there is no overlap of events. Because this is an example of mutually exclusive events, we can use this formula from the Addition Rule of Probability: Abby has a 1/7 chance of getting first place, and a 1/7 chance of getting second place. We can add these two probabilities together to find the probability of Abby getting first or second like this: 1/7 + 1/7 =2/7
Cumulative Frequency table example
Coach Bernard is starting his summer training for his football players. He wants to measure his players' progress as the training continues. He decides to record each player's 40-yard dash at the beginning of the training and then time the players again at the end of the training. The times are sorted into intervals.
Probability of Compound events
Finding the probability of more than one event happening together. With compound events, we will use the same formula to calculate the probability of each event occurring. To calculate the probability, we will use the formula: number of favorable outcomes over the number of total outcomes. Once we find the probability of each event occurring, we will multiply these probabilities together. Wendy and Kim are playing a new deluxe board game titled ED Portalopoly. It is Wendy's turn and she needs to roll the two dice and get both to be a six in order to land on the jackpot. What is the probability that Wendy will roll a six and then roll another six? To start this problem, we need to calculate each event separately. The total number of outcomes on the dice is six and the number of favorable outcomes is one, because there is only one six on a dice. So, the probability of Wendy getting a six on her first roll is 1 out of 6. Next, we need to calculate the probability of Wendy getting a six on her second roll. Since these dice are the same, the probability of getting a six on the second roll will also be 1 out of 6. Now that we know the probability of both events happening, we need to multiply these two fractions together. So 1/6 x 1/6 = 1/36. So, the probability of Wendy rolling both dice and getting a six on both is 1/36.
Complementary events
Complementary events are events that add together to equal a whole or one. For example, if the probability of it raining today were 2/5, what would the probability be of it not raining? Ex: Let's check back on Wendy and Kim playing everyone's favorite board game, ED Portalopoly. It is now Kim's turn to roll. If she rolls two dice that add together to equal eight, she will land in jail. Wendy is hoping that Kim lands in jail on this turn. What is the probability that Kim will not roll two dice whose sum is eight? Wendy needs to first calculate the probability that Kim will roll the two dice and get a sum of eight. To find this probability, she will use the formula: number of favorable outcomes over the number of total outcomes. Since Kim is rolling two dice and each dice has six sides, there are 36 total outcomes that she can get. To find the number of favorable outcomes, Wendy decides to list out possible ways that the two dice can add to eight. She knows that Kim could roll a 2 + 6, or a 3 + 5, or a 4 + 4, or a 5 + 3 or a 6 + 2. She can see that there are only five different ways to roll two dice whose sum will be eight. Wendy now knows that the probability of Kim rolling two dice and getting a sum of eight is 5 out of 36, or 5/36. Wendy also wants to know the probability of Kim not getting the two dice to add together to equal eight. These two events are complementary, because they will equal 1 or 36/36. Since the probability of Kim getting the sum of eight was 5/36, we can subtract to find the probability of her not getting the sum of eight. 36/36 - 5/36 = 31/36. The probability of Kim not rolling the two dice and getting the sum of eight is 31/36, so the chance that Kim does not land in jail on her next turn is 31/36.
Cumulative frequency and relative frequency
Cumulative frequency is the total number of times a specific event occurs within the time frame given. Relative frequency is the number of times a specific event occurs divided by the total number of events that occur. Your soccer team ended the season with a record of 15 wins and 3 losses. The cumulative frequency of your wins is 15 because that event occurred 15 times. The given time frame was the season The relative frequency of wins is 15 divided by 18, or 83%, because, out of the 18 total games (or events), your team won 15.
Univariate data
Data with only one variable (Ex: Weight of a bunch of dogs. The one variable is puppy weight)
Creating a histogram
Horizontal axis: Data Vertical Axis: Frequency of each piece of data Ex: Let's look at a histogram that deals more with numbers, rather than categories. Julianne wants to know how much time each person spends on each type of transportation. She realizes that some people use different types of transportation during the day. Julianne asks the same 50 people how many hours they spend on each type of transportation. After collecting the data, Julianne groups the information by types of transportation. Here is the data set for the number of hours each person spends riding the bus each week: 0, 5, 2, 3, 12, 10, 1, 1, 0, 4, 5, 2, 1, 0, 3, 4, 7, 1, 10, 12, 9, 7, 8, 1, 0, 2, 5, 7, 3, 5, 4, 1, 2, 3, 4, 2, 3, 4, 10, 1, 0, 3, 4, 2, 5, 7, 7, 9, 10, 6 Wow, that's a lot of numbers. It's a little too much to take in. If we just try to sit here and look at these numbers, we may be left with a headache but not much else. We can take these numbers and place them into a histogram to make the numbers easier to understand. Much better! The numbers at the bottom, along the horizontal axis, are the actual hours spent on a bus. The numbers at the side, along the vertical axis, are the number of times the hours appeared in the data set. Take 10 hours as an example. The bar above the 10 goes up to match the number 4 on the vertical axis. That means that 4 people said they ride the bus 10 hours a week. If you look at our original data set, you will see that 10 appears four times.
Conditional probability and independent probability
However, you can also look at conditional probability with two or more independent events, such as drawing a certain card from a deck and pulling a certain marble from a bag of marbles. The two events do not influence one another, but you can calculate the probability of one event, given one event has already occurred.
Postmodern Blackness
In her work Postmodern Blackness, hooks does want to acknowledge that the identities of black people are varied. There is not a uniform black experience, and no single experience of being female either. There is no one set description for what it is like to be a particular gender, class, ethnicity or sexuality. But she doesn't believe that systems of oppression are a thing of the past. For instance, she says the women we see portrayed in media images often have little to no control of how they are presented. Instead, those with money and power are in control and choose which images are promoted.
Rawls' thought experiment
In his thought experiment, you would imagine that you have a representative that will meet with the representatives of every other citizen in a society to come up with principles of justice. Those who are involved in this get-together are part of what Rawls describes as the original position, or an impartial point of view used to establish the principles of justice. They hold this point of view. What makes the original position of the participants so impartial and fair? Well, they've been given a strange characteristic. Each citizen's representative knows nothing about the actual social or economic standing of the person they are representing. They don't know your income, or whether you are the wealthy business owner, or the lower-income parent, for instance. They don't know your ethnicity, your gender, or your age, either. They don't even know what political or economic situation has been established in your society. This lack of information about the relative situation of those they represent is called the veil of ignorance. What do they know? They know you have a certain plan for life. They know you have an interest in having enough for yourself. They also know that not everyone in the society will get everything they want. Although resources are adequate for the society in our thought experiment, resources are not infinite. Part of developing principles of justice is considering how to share these resources fairly. Rawls argued that using this thought experiment highlights the importance of equal basic rights and access to opportunities like employment and education to help each person compete in the world.
Permutation and its formula
John is an avid card player. His favorite card game to play is poker. The best part about playing poker for him is the moment when the cards are dealt. John has always been curious about how many different ways he could organize his cards. He researched and found that a permutation is an arrangement of items or events in which order is important. To calculate permutations, we use the equation nPr, where n is the total number of choices and r is the amount of items being selected. To solve this equation, use the equation nPr = n! / (n - r)!.
Combination formula
The coach of the Wildcats now knows that he has to use the equation nCr = n!/(r!(n-r)!), where n represents the number of items and r represents the number of items being chosen at a time. Ex: The Jackson Wildcats play basketball in a highly competitive city district. There are eight teams in the district, and they all play each other once during the season. The coach of the Wildcats wants to know how many games will be played in the district this season. To calculate this amount, he will need to use a combination. The coach knows that there are eight teams, but the order the teams play each other does not matter. The coach of the Wildcats now knows that he has to use the equation nCr = n!/(r!(n-r)!), where n represents the number of items and r represents the number of items being chosen at a time. Using this equation, he must select two teams for each game from the eight teams in the district. So, the variable n would equal 8 and the variable r would equal 2. The equation would then look like 8 C 2 = 8!/2!(8-2)!.
Let's look at Sydney's data from the past year. Each number is the number of tomatoes sold each month for eight months. The data is ordered from least to greatest. 121, 121, 123, 124, 125, 127, 128, 132 Rounded to the nearest whole number, the mean of this data set is 125. Find the measures of central tendency, and make conclusions based on them
The mean can be used to get an overall idea or picture of the data set. Mean is best used for a data set with numbers that are closer together. Mean is not good for measuring the central tendency of data sets that contain outliers. Since this data set does not contain outliers, we can use the mean of this data set to make arguments and predictions. For example, we could make the argument that Sydney should buy seeds that will yield a minimum of 125 tomatoes, since on average she sold 125 tomatoes each month. Rounded to the nearest whole number, the median of this set of data is 125. The median can be used to get an idea of what values fall above the midpoint and what values fall below the midpoint. There is equal likelihood that the values in the data set will fall either above or below the median. Median is best used for a data set with numbers that have a few larger or smaller numbers and several numbers close together. One large or small number might skew the mean, but the median can often give you a better idea of the data. For example, if Sydney sold her tomatoes at the farmers market, and then a sudden storm caused the customers to leave, the sales for that day might skew her data. That's because it wasn't the tomatoes that caused fewer sales, it was the storm. In this case, the median would be a better indicator of central tendency. The mode is the easiest measure of central tendency to find; simply find the number that occurs the most in the data set. In this data set, the number that occurs the most is 121. Mode is a good way to analyze the frequency that certain numbers occur in a data set. If you are looking for the most popular option in a data set, mode is a good method to use. Let's look at the measures of central tendency that we have: Mean: 125Median: 125Mode: 121 Based on this data, Sydney would be best off buying at least enough tomato seeds to sell 125 tomatoes each month. We know this because the data set showed us that, on average, Sydney sold 125 tomatoes, while the mode was the smallest number in the data set. We can make the argument that Sydney will need to buy at least enough tomato seeds to sell 125 tomatoes each month. We can also predict that she will sell at least 125 tomatoes on average in the upcoming season.
This is a histogram that Julianne created. She gathered the data after asking how many people walked as part of their daily transportation system. Out of 50 people, 32 said they walked. Julianne then asked what distances each person walked daily. She grouped the distances and created this histogram. Interpret the histogram.
The mode of this data set is 0.15, since it appears 10 times. We can identify the mode of this data set because it is the distance that was selected the most from the people surveyed. 10 people said they walked 0.15 miles.
Let's look at Sydney's data from the past year. Each number is the number of tomatoes sold each month for eight months. The data is ordered from least to greatest. 121, 121, 123, 124, 125, 127, 128, 132 Rounded to the nearest whole number, the mean of this data set is 125.
This is a frequency table. It is a numerical way of showing the frequency distribution in a data set. If you draw a line after 125 and look at all of the observations in the data set between 121 and 125, you will see that most of the observations occur in this range. Therefore, the histogram and the frequency table both fit with our prediction of needing at least 125 tomatoes to sell each month. Another form of univariate data is called time series. This is a more advanced form of statistics, so we won't be covering it in depth here. Basically, a time series is a group of data compared to itself in the past. For example, Sydney could collect two or three seasons' worth of data and then compare each month's sales. In this instance, you would still have one variable, but you have several data sets to use for comparison.
Abby's team ranks first out of the other teams at the end of the swimming competition. The team goes out for pizza and ice cream afterward. There are 20 people on the team; 8 people order pizza, and 12 people order ice cream. Out of the team, 5 people ended up ordering both pizza and ice cream. What is the probability of a team member ordering pizza or ice cream, but not both?
This is an example of non-mutually exclusive events since some of the team members were able to order both ice cream and pizza. The probability of a team member ordering pizza is 8/20 since we were already given that information. The probability of a team member ordering ice cream is 12/20. First we can add those two probabilities together: 8/20 + 12/20 = 20/20 You've probably decided at this point there is something wrong, since there are only twenty people on the team. That's because at some point there is an overlap in the numbers. Remember, some people ordered both pizza and ice cream. We know from the problem that 5 people ordered both pizza and ice cream. We need to subtract that probability 5/20 from our problem like this: 20/20 - 5/20 = 15/20 Remember that probability is estimation or a prediction in this case. We are trying to predict whether or not a team member would actually order both, or one or the other. Therefore, we can only accurately say, that there were 5 people that ordered both. We can say if a teammate does not order both, there is a 75% chance he or she will order one or the other.
Let's look at Sydney's data from the past year. Each number is the number of tomatoes sold each month for eight months. The data is ordered from least to greatest. 121, 121, 123, 124, 125, 127, 128, 132 Rounded to the nearest whole number, the mean of this data set is 125. Create a box-and-whisker plot. Make conclusions
This is our data represented in a box-and-whisker plot. Notice that the box, or the interquartile range, is between 122 and 128. Interquartile range is best for when you are looking at a group of numbers and comparing them to the average, such as test scores or performance-based data like game scores. This gives us a visual representation on how many tomatoes Sydney will need for the next season. According to this, she will need to buy enough seeds to yield between 122 and 128 tomatoes each month, not far off from our prediction of 125.
Let's look at Sydney's data from the past year. Each number is the number of tomatoes sold each month for eight months. The data is ordered from least to greatest. 121, 121, 123, 124, 125, 127, 128, 132 Rounded to the nearest whole number, the mean of this data set is 125. Create a box-and-whisker plot. Create stem and leaf display.
This is our data represented in a stem-and-leaf display. We can see from this display that we will most likely not need more than 128 tomatoes per month. Once again, this fits with our original prediction of at least 125 tomatoes per month.
Let's try an example: Data for crime in a certain area was recorded over two years. The following table shows the occurrences of three different types of crime over a two-year period. Calculate the percentage increases over these two years
To calculate the percent increase, take each row individually and plug the numbers into the equation. Percent increase of robbery = ((37 - 33) / 33) * 100 = (4 / 33) * 100 = 0.12 * 100 = 12% Percent increase of murder = ((8 - 2) / 2) * 100 = (6 / 2) * 100 = 3 * 100 = 300% Percent increase of assault = ((16 - 15) / 15) * 100 = (1 / 15) * 100 = 0.07 * 100 = 7%
Probability of Simple Event
finding the probability of a single event occurring. When finding the probability of an event occurring, we will use the formula: number of favorable outcomes over the number of total outcomes. Ex: Let's look at an example of simple events. Sam owns a large fish store with many colors of fish. He keeps all of the fish in a large aquarium. In his main aquarium, he has 5 red fish, 6 blue fish, 14 white fish and 5 green fish. A customer comes into the store and wants to buy a blue fish to take home. What is the probability that Sam will reach in and scoop out a blue fish on his first scoop? To calculate this probability, Sam needs to find out the total number of fish in his aquarium. Sam needs to add the 5 red fish + 6 blue fish + 14 white fish + 5 green fish = 30 total fish. Sam must also know the number of favorable outcomes. In his tank, there are only six blue fish. So his number of favorable outcomes is six. The formula to calculate the probability is the number of favorable outcomes over the number of total outcomes. The number of favorable outcomes is six and the number of total outcomes is 30, so, the probability of Sam scooping a blue fish the first time is 6 out of 30. Remember, all fractions must be in simplest form, so 6 over 30 will reduce to 1 over 5. The probability that Sam will scoop out a blue fish on his first try is 1/5.
Conditional Probability
involves finding the probability of an event occurring based on a previous event already taking place For example, thinking about Walt and his gumballs, Walt had 3 red, 6 green, 8 blue and 2 orange gumballs. After first reaching in and selecting a red gumball, and WITHOUT REPLACING it, what is the probability that Walt's second draw will be another red gumball? Walt knows that the probability of the first gumball being red is 3/19. Now, he has one less red gumball and one less total gumballs. We can see that the probability of Walt's second gumball being another red would be 2/18. Remember, all fractions must be in simplest form. 2/18 would be simplified to 1/9. The conditional probability that Walt's second gumball will be red after first drawing a red and not replacing it is 1/9.
Expected value. Formulas?
number of successful outcomes expected in an experiment Brady is playing a game with his friends. He needs to roll a 4 to avoid any bad things from happening to him in the game. He has two chances to roll a 4. Before he rolls the die, his friends bet if he is going to make it. In Brady's case, the expected value is the probability that he will roll a 4 in two tries. Formulas: n*P; number of trials*Probability of rolling a given outcome in a single trial (For discrete random variable) The sum of (X times P(X)) for all values of a random variable X
conditional relative frequency
numbers are the ratio of a joint relative frequency and related marginal relative frequency Ex: For example, let's say you wanted to find the percentage of people that selected clown as a career, given those people are girls. You would then find the number of girls that selected clown and divide by the total number of girls in the survey. Travis also wants to know the number of people surveyed that were boys, given those people chose clown as a career. In this conditional relative frequency, we are basically asking the question, 'What percentage of people that selected clown are boys?'
binomial experiment
which is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. Before we talk more about Zoey's problem, let's talk about an experiment Gabe and his friends are conducting. Gabe and his friends are hanging out at the local arcade. His friends dare him to ask 20 girls for their phone numbers. The friends watch as Gabe has some trouble getting phone numbers from girls at the arcade. Gabe asks the twelfth girl for her number. He hasn't gotten any so far, so they bet that Gabe will be unsuccessful with this girl. There are some things to keep in mind when understanding binomial experiments. First, the outcomes must be independent. This means that the outcome of one trial cannot have any influence on another. With Gabe's experiment, we have to assume that when Gabe asks one girl for her phone number, her response does not influence the outcome of any other girl. If Gabe asks one girl for her phone number and she says no, and Gabe then asks her nearby friend who also says no, then it's possible that the first girl's response influenced the second girl's response. Only when the outcomes are independent can the experiment be considered binomial. Second, a binomial experiment must only have two possible outcomes. In the case of Gabe's experiment, he can only receive yes or no answers. If the girls answered 'maybe' or 'later', then this would not be considered a binomial experiment. Third, there are a fixed number of trials. Gabe and his friends said he would only ask 20 girls. If the bet, or experiment, was to ask as many girls as he could until he got a number, then this would not be a binomial experiment. Let's practice identifying them. Let's try another example. Zoey has a deck of cards; some of the cards are red, and some are black. Gabe bets that he can draw 5 black cards from the deck out of 20 draws. He tries, drawing 20 cards out of the deck. Fourteen of the cards are red and six of the cards are black. Is this a binomial experiment? We know that there are only two possible outcomes: red or black, and there are a fixed number of trials; Gabe knew he was only drawing 20 cards from the deck. The trick here is independence. Are the outcomes independent? Well, we don't really know. If Gabe drew a card from the deck, wrote down the color, and then put the card back before drawing another out; then yes, the outcomes are independent. However, if Gabe draws all 20 cards at once without replacing them, then the number of red cards and black cards in the deck decreases, and the outcome is dependent on the cards that have already been removed from the deck. Check out our other lessons on dependence and independence for more details!
A Vindication of the Rights of Women
written by Mary Wollstonecraft; A Vindication of the Rights of Women called for female equality, particularly in the area of education. Wollstonecraft dismissed the cultivation of traditional female virtues of submission and service and argued that women could not be good mothers, good wives and good household managers if they were not well-educated. She claimed that women were expected to spend too much time on maintaining their delicate appearance and gentle demeanor, sacrificing intelligence for beauty and becoming flower-like playthings for men. Wollstonecraft recognized that for many women of her time, raising a family would be their primary responsibility, but she insisted that a husband and wife whose relationship was founded on reason and equality would parent happier and more well-rounded children than in families governed by strict discipline and inequality between parents. To that end, she proposed a system of national education in which boys and girls would be educated together, and education would be open to all social classes.
