MATH - Ch. 5/6/7
Theoretically Method for Equally Likely Outcomes
1. Count the total of possible outcomes. 2. Among all the possible outcomes, count the number of ways the event of interest, A, can occur. 3. Determine the probability, P(A), form P(A)= number of ways A can occur/total number of outcomes
Quantifying Statistical Significance
1. If the probability of an observed difference occurring by chance is (0.05 or 5%, or 1 and 20) or less, the difference is statistically significant at 0.05 level. 2. If the probability of an observed difference occurring by chance (0.01 or 1%, or one and 100) or less, the difference is statistically significant at the 0.01 level.
Conditions for a Normal Distribution
1. Most data values are clustered near the mean, giving the distribution a well-defined peak. 2. Data values are spread evenly around the mean, making the distribution symmetric. 3. Larger deviations from the mean become increasingly rare, producing the tapering tails of the distribution. 4. Individual data values result from a combination of many different factors, such as genetic and environmental factors. ex: SAT/IQ Scores, Adult Heights, Sport Statistics
Relative Frequency Method
1. Repeat or observe a process many times and count the number of times the event of interest, A, occurs. 2. Estimate P(A) form P(A) = number of times A occured/ total number of observations
Determine the probability of the given complementary event. What is the probability of randomly selecting a month of the year and not getting a month that contains more than 8 letters?
11/12
Use the theoretical method to determine the probability of the following event. A randomly selected person has a birthday in July.
31/365
Probability Distribution
A probability distribution represents the probabilities of all possible events of interest. The sum of all the probabilities in a probability distribution must be 1.
Making a Probability Distribution
A probability distribution represents the probabilities of all possible events. Do the following to make a display of a probability distribution. 1. List all possible outcomes. Use a table or figure if it is helpful. 2. Identify outcomes that represent the same event. Find the probability of each event. 3. Make a table in which one column lists each event and another column lists each probability. The sum of all the probabilities must be 1.
The Law of Large Number
As a (independent) procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability = P(A).
HW; Use the theoretical method to determine the probability of the following outcome and event. State any assumptions made. Tossing two coins and getting either one head or two heads
Assuming that each coin is fair and is equally likely to land heads or tails, the probability is 3/4.
If the answer is yes ... If the answer is no...
(the probability is less than or equal to 0.05), then we say the difference is statistically significant at the 0.05 level. the observed difference is reasonably likely to have occurred by chance, so we say that it is not statistically significant at the 0.05 level.
HW Question; A roulette wheel has 38 numbers, with 18 odd numbers (black) and 18 even numbers (red), as well as 0 and 00 (which are green). If you bet $2 that the outcome is an odd number, the probability of losing the $2 is 20/38 and the probability of winning $4 (for a net gain of only $2, given you already paid $2) is 18/38. Complete parts (a) and (b) below. a. If a player bets $2 that the outcome is an odd number, what is the player's expected value? _______ dollars. (Round to the nearest cent as needed.) b. Is the best option to bet $2 on odd or to not bet? Why?
-2(20 ÷ 38) + 2(18 ÷ 38)= −.11; Not betting is best because it has the highest expected value.
nth percentile
Is the smallest value in the set with property that n% of the data values are less than or equal to it. A data value that lies between two percentiles is said to lie in the lower percentile.
What is a best fit line? How is a best fit line useful?
It is a line that lies closer to the data points than any other possible line; The best-fit line (or regression line) on a scatterplot is a line that lies closer to the data points than any other possible line (according to a standard statistical measure of closeness). It is useful to make predictions within the bounds of the data points; Best-fit lines are useful in making predictions within the bounds of the data points because the data points are used in constructing the best-fit line; a best-fit line lies closer to the data points than any other straight line that could be drawn through the data.
HW Question; ; State whether the difference between what occurred and what you would have expected by chance is statistically significant. In a clinical trial of a new drug intended to treat allergies, 3 of the 80 subjects in the treatment group experienced headaches, and 9 of the 160 subjects in the control group experienced headaches. Is the difference int he percentages between the two groups statistically significant?
No
HW Question; One experiment conducted a clinical trial of a method for gender selection. According to this experiment, 340 babies had been born to parents using the new method to increase the probability of conceiving a girl, and 184 of those babies were girls. Discuss whether these results appear to be statistically significant. Do these results appear to be statistically significant?
No
HW Question; One experiment conducted a clinical trial of a method for gender selection. According to this experiment, 350 babies had been born to parents using the new method to increase the probability of conceiving a girl, and 189 of those babies were girls. Discuss whether these results appear to be statistically significant. Do these results appear to be statistically significant?
No
HW Question; State whether the difference between what occurred and what you would have expected by chance is statistically significant. In a clinical trial of a new drug intended to treat allergies, 5 of the 80 subjects in the treatment group experienced headaches, and 12 of the 160 subjects in the control group experienced headaches. Is the difference int he percentages between the two groups statistically significant?
No
HW Question; State whether the difference between what occurred and what you would have expected by chance is statistically significant. In conducting a survey of adults, a pollster claims that he randomly selected 30 subjects and 16 of them were men. Is the difference between what occurred and what is expected by chance statistically significant?
No
Question; Determine whether the following data set is likely to be normally distributed. Explain the reasoning. The measured systolic blood pressure of randomly selected adult males.
Normally distributed. Systolic blood pressure levels vary above and below the mean by similar amounts. The distribution has one peak and is symmetric. Systolic blood pressure results from many factors.
For the following pair of variables, state whether you believe the two variables are correlated. If you believe they are correlated, state whether the correlation is positive or negative. Explain your reasoning. The height of students and the lengths of their pants.
Positive correlation because taller people tend to have longer pants A correlation exists between two variables when higher values of one variable consistently go with higher values of another variable or when higher values of one variable consistently go with lower values of another variable.
Define Positive Correlation.
Positive correlation means that both variables tend to increase (or decrease) together. An example might be shoe size and height.
Expressing Probability
Probability is represented by P(E) which means probability (P) of a certain event (E) occurring. R = rain, and a 40% probability of rain would be expressed as P(R) = .40
Relative Frequency Technique
Relative frequency technique is based on observations or experiments. ex: Based on statistical data, the chance of having the championship team coming from the Eastern Conference of a certain basketball league is about 1 in 10. This is a relative frequency probability because it is based on basketball data from previous years.
Let A denote the event of getting a female when you randomly select a fellow student in your statistics class. Let B denote the event of getting a female when you randomly select a fellow student in your psychology class. Are events A and B independent or dependent?
Since the student that is chosen from the statistics class _does not affect_ the probability of choosing a female from the psychology class, the two events are _independent_.
HW Question; Give an example in which the same event can occur through two or more outcomes.
Suppose you roll a fair, six-sided die. The possible outcomes are rolling the number 1, 2, 3, 4, 5, or 6. The event of rolling an even number will occur with the three outcomes 2, 4, and 6.
When does the Central Limit Theorem apply?
The Central Limit Theorem applies to variables that follow any distribution; however, the sample size must be suitably large. A common threshold is n>30.
What does the area under the normal distribution curve represent? What is the total area under the normal distribution curve?
The area that lies under the normal distribution curve corresponding to a range of values on the horizontal axis is the total relative frequency of those values. Because the total relative frequency for all values must be 1 (100%), the total area under the normal distribution curve must equal 1 (100%).; The relative frequency for any range of data values is the area under the curve covering that range of values. Also, the total relative frequency for any data set must be 1, or 100%.
What is the law of large numbers? Can the law be applied to a single observation or experiment?
The law of large numbers states that if a process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). The larger the number of trials, the closer the proportion should be to P(A). It does not apply to a single trial (observation or experiment), or even to small numbers of trials, but only to a large number of trails. The law only applies to the long term behavior of the frequency of outcomes after many repeated trials, and because of this, offers no information concerning the outcomes of an individual trial.
For the description below, state the correlation clearly. (For example, state that "there is a positive correlation between variable A and variable B.") Then state whether the correlation is most likely due to coincidence, a common underlying cause, or a direct cause. Explain your answer. In one state, the number of unregistered handguns steadily increased over the past several years, and the crime rate increased as well. What is the correlation? Is the correlation most likely due to coincidence, a common underlying cause, or a direct cause?
The number of unregistered handguns increase and the crime rate increases. Since both variables increase together, there is a positive correlation between the two variables. The correlation is most likely due to a common underlying cause. Many crimes are committed with handguns that are not registered.; As a population increases, both the number of unregistered handguns and the crime rate are likely to increase, so the correlation is most likely due to a common underlying cause.
A bag contains 5 red candies, 15 blue candies, and 10 yellow candies. What is the probability of drawing a red candy? A blue candy? A yellow candy? Something besides a yellow candy? The probability of drawing a red candy is
The probability of drawing a red candy is 0.17. The probability of drawing a blue candy is 0.50. The probability of drawing a yellow candy is 0.33. The probability of drawing something besides a yellow candy is 0.67.
This exercise involves a complementary event. Find the probability of the given event. Assume that the deck is fair. Not drawing a black card from a standard deck of 52 playing cards.
The probability of not drawing a black card is 1/2 .
Use the theoretical method to determine the probability of the given outcome or event. Assume that the die is fair. Rolling a single six-sided die and getting an even number (2, 4, or 6).
The probability rolling a single six-sided die and getting an even number (2, 4, or 6) is one half 1/2.
After recording the forecasts of your local weatherman for 30 days, you conclude that he gave a correct forecast 9 times. What is the probability that his next forecast will be correct? Use the relative frequency method to estimate the probability.
The probability that the next forecast will be correct is 0.3 (9/30).
Halfway through the season, a basketball player has hit 81% of her free throws. What is the probability that her next free throw will be successful? Use the relative frequency method to estimate the probability.
The probability that the player's next free throw will be successful is .81
This is true for the possible range of values for P(A)
The range of possible values for P(A) is from 0 to 1 (inclusive), with 0 meaning there is no chance that event A will occur and 1 meaning it is certain that event A will occur.
HW Question; The numbers 5, 17, 18, 27, 36, and 41 were drawn in the last lottery; they should not be bet on in the next lottery because they are now less likely to occur.
The statement does not make sense because lottery drawings are independent events, which means that the outcome of one does not affect the probabilities of the other. Because winning lottery numbers are chosen randomly, the probability of winning with the last lottery's winning numbers is the same as the probability of winning with any other set of numbers.
HW Question; Determine whether the statement makes sense or does not make sense, and explain your reasoning. I found a strong negative correlation for data relating the percentage of people in various countries who are literate and the percentage who are undernourished. I concluded that an increase in literacy causes a decrease in undernourishment.
The statement does not make sense. Correlation is not necessarily causation. Establishing that one thing causes another is extremely difficult, even if there is a strong correlation between these things. For example, as the air temperature increases, there is an increase in the number of people stung by jellyfish at the beach. This does not mean that an increase in air temperature causes more people to be stung. It might mean that because it is hotter, more people go into the water. With an increased number of swimmers, more people are likely to be stung. Examine the variables given in the problem and determine if you can conclude, in this case, that changes in one of the given variables cause changes in the other.
Determine whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly. The scatterplot showed all the data points following a nearly straight diagonal line, but only a weak correlation between the two variables being plotted.
The statement does not make sense. The data points following a nearly straight diagonal line would indicate a very strong correlation between the two variables. Review how the shape of a scatterplot relates to the correlation between the variables. The scatterplot for weakly positively correlated variables will show points very loosely scattered around the plot, but following a vague increasing trend. The scatterplot for strongly positively correlated variables will show points scattered about a line with a positive slope. The scatterplot for perfectly positively correlated variables will show points that all lie on a line with a positive slope. The scatterplot for variables with no correlation will not have any discernable trend. Similar statements hold true for negatively correlated variables, except the slope of the trend lines will be negative rather than positive.
Determine whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly. The two variables I studied showed such a strong correlation that they had a correlation coefficient of r=1.50.
The statement does not make sense. The value of the correlation coefficient ranges from −1 to 1, so having a value of r=1.50 is not possible.
The difference between theoretical, relative frequency, and subjective techniques for finding probabilities?
The theoretical technique is based on the assumption that all outcomes are equally likely. It is determined by dividing the number of ways an event can occur by the total number of possible outcomes. The relative frequency technique is based on observations or experiments. It is the relative frequency of the event of interest. The subjective technique is an estimate based on experience or intuition.
Theoretical Techniques
The theoretical technique is based on the assumption that all outcomes are equally likely. ex: The probability of rolling a 3 on a single die is 1/6. This is a theoretical probability because it is based on assuming that a fair die is equally likely to land on any of its 6 sides.
Statistics students find that as they spend more time studying, their test scores are higher. What is the correlation? For the description below, state the correlation clearly. (For example, state that "there is a positive correlation between variable A and variable B.") Then state whether the correlation is most likely due to coincidence, a common underlying cause, or a direct cause. Explain your answer. Statistics students find that as they spend more time studying, their test scores are higher. What is the correlation? Is the correlation most likely due to coincidence, a common underlying cause, or a direct cause?
The times studying increase and the test scores increase. Since both variables increase together, there is a positive correlation between the two variables. As students study more, they gain a better understanding of the subject and their test scores are likely to be higher, so the correlation is most likely due to a direct cause. Increasing studying times results in an increase in test scores.
normal
The word normal does have a special meaning in statistics. It refers to a specific category of distributions that are symmetric and bell-shaped with a single peak. The peak corresponds to the mean, median, and mode of such a distribution. The variation of the distribution can be characterized by its standard deviation.
It has been found that as gas prices increase, the distances vehicles are driven tend to get shorter. What is the correlation? Is the correlation most likely due to coincidence, a common underlying cause, or a direct cause?
There is negative correlation between gas prices and the distances vehicles are driven; The gas prices increase and the distances vehicles are driven decrease. When one variable increases, the other decreases, so there is a negative correlation between the two variables. The correlation is most likely due to a direct cause. As gas prices increase by large amounts, people can't afford to drive as much, so they cut costs by driving less; As gas prices increase, people cut costs by driving less, so the correlation is most likely due to a direct cause. Higher gas prices result in less driving.
It has been found that as the number of traffic lights increases, the number of car crashes also increases. What is the correlation? Is the correlation most likely due to coincidence, a common underlying cause, or a direct cause?
There is positive correlation between the number of traffic lights and the number of car crashes; The number of traffic lights increases and the number of car crashes increases. Since both variables increase together, there is a positive correlation between the two variables. The correlation is most likely due to a common underlying cause, such as the general increase in the number of cars and traffic.
Question; State, with an explanation, whether you would expect the following data set to be normally distributed. The delay in departure of trains from a station (note that trains, buses, and airplanes cannot leave early)
This data set is not normally distributed. There is no reason to assume that the mean, median, and mode of this distribution are centered at a central peak in a symmetric, bell-shaped distribution. It is possible that trains never experience delay or frequently encounter very above average delays at this particular station. If the answer is incorrect; Carefully review the definition and qualifications for a normal distribution. Consider whether this scenario will behave irregularly or if values will indeed cluster toward the mean and become less common farther from the mean
Independent Events (AND Probability for Independent Events)
Two events are independent if the occurrence of one event does not affect the probability of the other event. Since the students are picked from different classes, the two events are independent. P(A and B) = P(A) x P(B) P(A and B and C) = P(A) x P(B) x P(C)
HW Question; Assume that boys and girls are equally likely and that the gender of a child is independent of the gender of any brothers or sisters. If a couple already has three girls, find the probability of getting a girl when their fourth baby is born.
Two events are independent if the outcome of one event does not affect the probability of the other event. Since the gender of a child is independent of the gender of any brothers or sisters, the probability of a couple with three girls having another girl is the same as the simple probability of having a girl. It is stated that boys and girls are equally likely, so this probability is 1/2 or 0.5.
Either/Or Probability for Non-overlapping Events
Two events are non-overlapping if they cannot occur at the same time. If A and B are non-overlapping events, the probability that either A or B occurs is P(A or B) = P(A) + P(B) This principle can be extended to any number of non-overlapping events. For example, the possibility that either event A, event B, or event C occurs is P(A or B or C) = P(A) + P(B) + P(C) provided that A, B, and C are all non-overlapping events
Unusual values
Values that are more than 2 standard deviations away from the mean (@ 3rd Standard deviation; that 5%)
What do we mean when we say that a result is statistically significant?
When the difference between what is observed and what is expected seems unlikely to be explained by chance alone. (it is unlikely to have happened by chance).
HW Question; One experiment conducted a clinical trial of a method for gender selection. According to this experiment, 325 babies had been born to parents using the new method to increase the probability of conceiving a girl, and 293 of those babies were girls. Discuss whether these results appear to be statistically significant. Do these results appear to be statistically significant?
Yes
HW Question; State whether the difference between what occurred and what you would have expected by chance is statistically significant. In 500 tosses of a coin, you observe 400 tails.
Yes
HW; As long as all outcomes are equally likely, the theoretical method can be used to calculate the probability. Are all outcomes equally likely?
Yes
Question; In 120 rolls of a six-sided die, the outcome of 3 appears three times. State whether the difference between what occurred and what you would have expected by chance is statistically significant. Is the difference between what occurred and what is expected by chance statistically significant?
Yes
The event not A is called _________________ of the event A; "not" is often designated by a bar, so A means not A.
complement
Correlation
exists between two variables when higher values of one variables are consistently associated with higher values of another variable or when higher values of one variables are consistently associated with lower values of another variables.
A normal distribution
is a symmetric, bell-shaped distribution with a single peak. The peak corresponds to the mean, median, and mode of the distribution. The variations of the distribution can be characterized by its standard deviation.
A standard score
is the number of standard deviations a data value lies above or below the mean. ex: The standard score of the mean is z = 0, because it is 0 standard deviation from the mean; 1.5 standard deviations above the mean is z= 1.5, and 2.4 standard deviations below the mean is z= -2.4.
Question; The histogram ____ close to normal, since it is bell dash shaped and symmetric about a single mode _____-_________ _____ _____________ _______ __ _________ _____. This variable ________ have a normal distribution, since this variable can be expected to have many values near the ________ of the distribution and to be __________.
is; bell-shaped and symmetric about a single mode. should; center; symmetric
P(A) means
the probability that event A will occur. P(A)=number of times A occured / total number of observations. OK
P(not A) means
the probability that event A will not occur.
negative correlation
two variables move in opposite directions (one increasing while the other is decreasing)
HW Question; Football teams have the option of trying to score either 1 or 2 extra points after a touchdown. They get 1 point by kicking the ball through the goal posts or 2 points by running or passing the ball across the goal line. For a recent year, 1-point kicks were successful 91% of the time, while 2-point attempts were successful only 37% of the time. In either case, failure means zero points. Calculate the expected values of the 1-point and 2-point attempts. Can you think of any circumstances in which a team should make a decision different from what the expected values suggest? Explain. Calculate the expected value of the 1-point attempt. Calculate the expected value of the 2-point attempt. What are the most appropriate circumstances in which a team should make a decision different from what the expected values suggest? Explain.
(1 × 0.97) + (0 × 0.03) = .91 (2 × 0.37) + (0 × 0.63) = .74 A team has better experience in running or passing the ball across the goal line, or a team is behind by more than 1 point and the game is almost over.
Central Limit Theorem
1. The distribution of means will be approximately a normal distribution. 2. The mean of the distribution of means approaches the population mean, μ. 3. The standard deviation of the distribution of means approaches σ/n, where σ is the standard deviation of the population.
What is a correlation?
A correlation exists between two variables when higher values of one variable consistently go with higher or lower values of another variable. Ex: Three examples of correlations are amount of smoking and lung cancer, height and weight of people, price of a good and demand of the good. Next Question
Define No Correlation
No correlation exists when there is no apparent relationship between the two variables. An example might be hair color and weight.
Distinguish between an outcome and an event in probability.
Outcomes are the most basic possible results of observations or experiments. An event consists of one or more outcomes that share a property of interest.
If the probability of an event A is P(A), then the probability that event A does not occur is P(not A). Because the event must either occur or not occur, one can write the following equations. The event "not A", often denoted A, is called the complement of A.
P(A) + P (not A) = 1 or P(not A) = P (__A__) = 1- P(A)
Question; The figure to the right shows a histogram for the body temperatures (in °F) of a sample of 498 adults. Is this distribution close to normal? For the population of all adults, should body temperature have a normal distribution? Why or why not? Is this distribution close to normal? For the population of all adults, should body temperature have a normal distribution? Why or why not?
The histogram is symmetric around a single peak and bell-shaped, so the distribution is close to normal. Yes, because body temperature is a human trait determined by many genetic and environmental factors. The values for this variable should cluster near a mean and become less common farther from the mean, giving the distribution a bell shape.
An experiment consists of drawing 1 card from a standard 52-card deck. What is the probability of drawing a diamond?
The probability of drawing a diamond is 1/4 .
Key Idea
The relative frequency for any range of data values is the area under the curve covering that range of values.
Standard Score Formula
The standard score for a particular data value is given by z = data value−mean/standard deviation.
HW Question; The probability of rolling a die and getting an even number is 1/2, and the probability of getting an odd number is 1/2, so the probability of getting a number that is even or odd is 1/2+1/2=1.
The statement makes sense. Since a number cannot be even and odd, it is a valid application of the either/or rule for non-overlapping events.
Explain how to make a table of a probability distribution.
To make a table of a probability distribution, list all possible outcomes, identify the outcomes that represent the same event, and then find the probability of each event. These are the steps necessary to make a table of probability distribution.
Either/Or Probability for Overlapping Events
Two events A and B are overlapping if they can occur together. For overlapping events, the probability that either A or B occur is P(A or B) = P(A) + P(B) - P(A and B) The last term, P(A and B), corrects for the double counting of events in which A and B both occur together.
Dependent (AND Probability for Independent Events)
Two events are dependent if the outcomes of one event does affect the probability of the other event. The probability that dependent events A and B occur together is P(A and B) = P(A) x P(B given A) where "P(B given A)" means the probability of event B given the occurrence of event A. P(A and B and C) = P(A) x P(B given A) x P(C given A and B).
A pollster randomly selects an adult for a survey. Let M denote the event of getting a male, and let R denote the event of getting a Republican. Are events M and R overlapping?
Two events are overlapping if they can occur together. The pollster could select an adult who is male and Republican.
postive correlation
a relationship between two variables in which both variables either increase or decrease together
correlation coefficient
a statistical index of the relationship between two things (from -1 to +1)
multiple regression
allows the calculation of a best-fit equation that represents the best fit between one response variable (such as price) and a combination of two or more explanatory variables (such as weight and color). The coefficient of determinism, R^2, tells us the proportion of the scatter in the data accounted for by the best-fit-equation.
Expected Values
expected values = (value of event 1) x (probability of event 1) + (value of event 2) x (probability of event 2).
Two events are ______________ if the outcome of one event does not affect the probability of the other event.
independent
Event
is a collection of one or more outcomes that share a property of interest. For example, if you toss two coins and count the number of heads, the outcomes HT and TH both represents the same event of 1 head (and 1 tail).
The standard deviation of a normal distribution is
larger if its values are more spread out.
best fit line
the line that most closely approximates the data in a scatter plot
nonlinear relationship
the two variables are related, but the relationship results in a scatter diagram that does not follow a straight-line pattern
no correlation
there does not appear to be a relationship between two sets of data (variables)
Scatterplot
(or scatter diagram) is a graph in which paired sample data values are plotted as points with values of one variable on the horizontal axis and values of the other variable on the vertical axis. If we suspect that one variable depends on the other, we plot explanatory variable (which helps explain the values of the other variables) on the horizontal axis and the response variable (which) on the vertical axis.
Determine the probability of the given complementary event. What is the probability that a 41% free-throw shooter will miss her next free throw? The probability that a 41% free throw shooter will miss her next free throw is ...
0.59
What is the probability that a 0.320 hitter in baseball will not get a hit on his next at-bat? The probability that a 0.320 hitter in baseball will not get a hit on his next at-bat is
0.680
The 68-95-99.7 Rule for a Normal Distribution
1. About 68% (more precisely, 63.3%), or just over 2/3rds, of the data values fall within 1 standard deviation of the mean. 2. About 95% (more precisely, 95.4%) of the data fall within 2 standard deviation of the mean. 3. About 99.7% of the data fall within 3 standard deviation of the mean.
Statistical Significance; Suppose you toss a coin 100 times. Should you expect to get exactly 50 heads? Why or why not?
Due to small deviations by chance, you would get exactly 50 heads only about 8% of the time. However, if the coin is fair, you should expect to get a result that is fairly close to 50 heads.
What is an expected value? How is the expected value EV of two events computed? Should we always expect to get the expected value?
Expected value is the estimated gain or loss of partaking in an event many times. It's computed like this: EV = (event 1 value) x (event 1 probability) + (event 2 value) x (event 1 probability). We should not always expect to get the expected value because expected value is calculated with the assumption that the law of large numbers will come into play.; Expected value is only expected if the events considered can occur many times, making the outcome predictable.
Overlapping Event
For overlapping events, the probability that either A or B occurs is P(A or B)=P(A)+P(B)−P(A and B).
Non-Overlapping Events
If A and B are non-overlapping events, the probability that either A or B occurs is P(A or B)=P(A)+P(B).
Define Negative Correlation.
Negative correlation means that two variables tend to change in opposite directions, with one increasing while the other decreases. An example might be age and vision.
For the following pair of variables, state whether you believe the two variables are correlated. If you believe they are correlated, state whether the correlation is positive or negative. Explain your reasoning. The heights and weight of 50 randomly selected males between the ages of 10 and 21.
P. C. because taller men tend to weigh more.
Use the theoretical method to determine the probability of the outcome or event given below. The next president of the United States was born on Wednesday, Friday, or Saturday.
The probability of the given event is 3/7.
Outcomes
are the most basic possible results of observations or experiments. For example, if you toss two coins, one possible outcome is HT and another possible outcome.
Subjective Technique
subjective technique is an estimate based on experience or intuition. ex: My teacher assures me that he is certain that my SAT scores will be the highest for the entire country. This is a subjective probability based on the opinion of the teacher.