Independent Random Variable Combination - AP Statistics
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A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
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Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from 
because each trial is independent of the others.
Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from
Which of the following represents a continuous variable?
Which of the following represents a continuous variable?
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A continuous variable may assume an infinite number of different values between two given points while a discrete variable may only assume a specific number of specific values between two given points. Given this information, it becomes apparent that the concentration of DNA in a liquid sample and the weight of footballs are both continuous variables.
A continuous variable may assume an infinite number of different values between two given points while a discrete variable may only assume a specific number of specific values between two given points. Given this information, it becomes apparent that the concentration of DNA in a liquid sample and the weight of footballs are both continuous variables.
A group of student researchers want to test the impact of temperature and lighting on the amount of food laboratory rats will eat.
To do this, they will first vary the room temperature (Hot or Cold) and see how much the lab rats eat at each temperature.
They will then vary the lighting (On or Off) and see how much the lab rats eat during each lighting state.
Which of the following is the Dependant Variable?
A group of student researchers want to test the impact of temperature and lighting on the amount of food laboratory rats will eat.
To do this, they will first vary the room temperature (Hot or Cold) and see how much the lab rats eat at each temperature.
They will then vary the lighting (On or Off) and see how much the lab rats eat during each lighting state.
Which of the following is the Dependant Variable?
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A Dependent variable is what changes based on the independent, manipulated variable.
The Independent variables are the Room Temperature and the Lighting state, which the researchers were manipulating themselves.
The Dependent variable is the Mouse Food Intake, which is the variable that changed based on how the independent variables changed.
A Dependent variable is what changes based on the independent, manipulated variable.
The Independent variables are the Room Temperature and the Lighting state, which the researchers were manipulating themselves.
The Dependent variable is the Mouse Food Intake, which is the variable that changed based on how the independent variables changed.
A hair care company is testing a new shampoo formula for its ability to keep hair clean for longer thanks to a new chemical compound XYZ that should limit oil production.
To test the appropriate strength of the shampoo, they are testing several concentrations of chemical XYZ to and measuring the amount of oil on the participant's scalp after 24 hours. The color of the shampoo will be randomly changed to keep the participants from knowing what shampoo is which strength.
Which of the following is the Dependant variable?
A hair care company is testing a new shampoo formula for its ability to keep hair clean for longer thanks to a new chemical compound XYZ that should limit oil production.
To test the appropriate strength of the shampoo, they are testing several concentrations of chemical XYZ to and measuring the amount of oil on the participant's scalp after 24 hours. The color of the shampoo will be randomly changed to keep the participants from knowing what shampoo is which strength.
Which of the following is the Dependant variable?
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The Independent variable is what the experimenters are directly manipulating-- here, that is the strength of the Chemical XYZ.
The Dependent variable is the variable that changes based on the Independent variable-- in this case, it is the amount of oil on participants' scalps.
The Independent variable is what the experimenters are directly manipulating-- here, that is the strength of the Chemical XYZ.
The Dependent variable is the variable that changes based on the Independent variable-- in this case, it is the amount of oil on participants' scalps.
In a standard deck of cards, with replacement, what is the probability of drawing two Ace of Hearts?
In a standard deck of cards, with replacement, what is the probability of drawing two Ace of Hearts?
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Probability of drawing first ace of hearts: 
Probability of drawing second ace of hearts: 
Multiply both probabilities by each other:
Probability of drawing first ace of hearts:
Probability of drawing second ace of hearts:
Multiply both probabilities by each other:
A dice is rolled five times in a row. Each time, a 6 is rolled. What is the probability that a 6 will be rolled on the sixth roll? Assume that the dice is fair and has six sides.
A dice is rolled five times in a row. Each time, a 6 is rolled. What is the probability that a 6 will be rolled on the sixth roll? Assume that the dice is fair and has six sides.
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Regardless of the previous results of the dice rolls, the probability of each individual dice roll remains the same. Each time the dice is rolled, there is a 1/6 chance that it will land on a specific number. Therefore, there is a 1/6 chance that the dice will roll a 6 on the sixth roll.
Regardless of the previous results of the dice rolls, the probability of each individual dice roll remains the same. Each time the dice is rolled, there is a 1/6 chance that it will land on a specific number. Therefore, there is a 1/6 chance that the dice will roll a 6 on the sixth roll.
What is the probability that a person will flip a coin and land on heads eight times in a row? Assume that the coin is fair and can either land on heads or tails.
What is the probability that a person will flip a coin and land on heads eight times in a row? Assume that the coin is fair and can either land on heads or tails.
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In this problem, it is important to identify that flipping a coin is an independent event. One coin flip does not affect the next coin flip. In each flip, there is a 50% chance of landing on heads and a 50% chance of landing on tails. By multiplying the probabilities with each other eight times, we get the following:
In this problem, it is important to identify that flipping a coin is an independent event. One coin flip does not affect the next coin flip. In each flip, there is a 50% chance of landing on heads and a 50% chance of landing on tails. By multiplying the probabilities with each other eight times, we get the following:
A dice is rolled four times in a row. What is the probability that every one of the four rolls will result in a result greater than 2? Assume the dice is fair and has six sides.
A dice is rolled four times in a row. What is the probability that every one of the four rolls will result in a result greater than 2? Assume the dice is fair and has six sides.
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In this problem, each roll of the dice is independent from the previous roll. The probability of obtaining a result of greater than 2 on a single roll is 4/6 or 2/3 (since 3, 4, 5, and 6 are all greater than 2). In order to find the probability that this occurs four times in a row, we must multiply the following:
In this problem, each roll of the dice is independent from the previous roll. The probability of obtaining a result of greater than 2 on a single roll is 4/6 or 2/3 (since 3, 4, 5, and 6 are all greater than 2). In order to find the probability that this occurs four times in a row, we must multiply the following:
A coin is tossed 100 times and each time, the coin lands on tails. What is the expected probability of the coin landing on tails if it is rolled an infinite number of times? Assume that the coin is fair and has two sides - heads and tails.
A coin is tossed 100 times and each time, the coin lands on tails. What is the expected probability of the coin landing on tails if it is rolled an infinite number of times? Assume that the coin is fair and has two sides - heads and tails.
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According to the Law of Large Numbers, the probability of the coin landing on tails after being rolled an infinite number of times is equal to the probability of the coin landing on tails in an individual roll: 50%.
According to the Law of Large Numbers, the probability of the coin landing on tails after being rolled an infinite number of times is equal to the probability of the coin landing on tails in an individual roll: 50%.
Four people are playing a card game and each of them has their own 52-card deck that has been randomly shuffled.
What is the probability that all four of the individuals will draw a face card at random?
Four people are playing a card game and each of them has their own 52-card deck that has been randomly shuffled.
What is the probability that all four of the individuals will draw a face card at random?
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Since each deck of cards is separate and only one card is being drawn from each deck, the events are independent of each other. The probability of drawing a single face card (Jacks, Queens, or Kings) from a deck of cards is 12/52. We can multiply the probabilities as follows:
Since each deck of cards is separate and only one card is being drawn from each deck, the events are independent of each other. The probability of drawing a single face card (Jacks, Queens, or Kings) from a deck of cards is 12/52. We can multiply the probabilities as follows:
A student is conducting different experiments that include the rain fall totals in his town, the change in temperature, the tidal levels on the beach, and the humidity. These will all be measured over the next couple of months to figure out the current trends. What would the independent event be in these experiments?
A student is conducting different experiments that include the rain fall totals in his town, the change in temperature, the tidal levels on the beach, and the humidity. These will all be measured over the next couple of months to figure out the current trends. What would the independent event be in these experiments?
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Time is the independent variable as time is not affected by the other variables but the other variables all change as time lapses.
Time is the independent variable as time is not affected by the other variables but the other variables all change as time lapses.
If 
is a random variable with a mean of 
and standard deviation of 
, what is the mean and standard deviation of 
?
If
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Remember how the mean and standard deviation of a random variable are affected when it is multiplied by a constant.
Remember how the mean and standard deviation of a random variable are affected when it is multiplied by a constant.
If you have ten independent random variables 
, normally distributed with mean 
and variance 
, what is the distribution of the average of the random variables, 
If you have ten independent random variables
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Any linear combination of independent random variables is also normally distributed with the mean and variance depending on the weights on the random variables. The mean is additive in the sense that
Each 
is 
, so the sum is equal to zero.
This means the sum of the average
is 
.
The variance satisfies
because of independence.
This means that the average is normally distributed with mean 
and variance 
.
Any linear combination of independent random variables is also normally distributed with the mean and variance depending on the weights on the random variables. The mean is additive in the sense that
Each
This means the sum of the average
The variance satisfies
This means that the average is normally distributed with mean
Suppose you have three independent normally distributed random variables, 
, such that
has mean 
and variance 
,
has mean 
and variance 
,
has mean 
and variance 
.
What is the probability that the sum, 
, is less than 
?
Suppose you have three independent normally distributed random variables,
What is the probability that the sum,
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There is a relatively simple way of doing this problem. The sum of any set of independent normal random variables is also distributed normally. So 
has a normal distribution. Now we can compute the mean and variance. The mean is additive:
Variance is also additive in some sense, when the random variables are independent:
Thus, 
is normally distributed with mean 
and variance 
.
This sum is a standard normal distribution.
The chance that 
is thus 
, if we use a normal table.
There is a relatively simple way of doing this problem. The sum of any set of independent normal random variables is also distributed normally. So
Variance is also additive in some sense, when the random variables are independent:
Thus,
This sum is a standard normal distribution.
The chance that
An experiment is conducted on the watermelons that were grown on a small farm. They want to compare the average weight of the melons grown this year to the average weight of last year's melons. Find the mean of this year's watermelons using the following weights:
An experiment is conducted on the watermelons that were grown on a small farm. They want to compare the average weight of the melons grown this year to the average weight of last year's melons. Find the mean of this year's watermelons using the following weights:
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To find the mean you sum up all of your values then divide by the total amount of values. The total sum of the weights is 
and there are 10 melons. 
To find the mean you sum up all of your values then divide by the total amount of values. The total sum of the weights is
A high school calculus exam is administered to a group of students. Upon grading the exam, it was found that the mean score was 95 with a standard deviation of 12. If one student's z score is 1.10, what is the score that she received on her test?
A high school calculus exam is administered to a group of students. Upon grading the exam, it was found that the mean score was 95 with a standard deviation of 12. If one student's z score is 1.10, what is the score that she received on her test?
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The z-score equation is given as: z = (X - μ) / σ, where X is the value of the element, μ is the mean of the population, and σ is the standard deviation. To solve for the student's test score (X):
X = ( z * σ) + 95 = ( 1.10 * 12) + 95 = 108.2.
The z-score equation is given as: z = (X - μ) / σ, where X is the value of the element, μ is the mean of the population, and σ is the standard deviation. To solve for the student's test score (X):
X = ( z * σ) + 95 = ( 1.10 * 12) + 95 = 108.2.
and 
are independent random variables. If 
has a mean of 
and standard deviation of 
while variable 
has a mean of 
and a standard deviation of 
, what are the mean and standard deviation of 
?
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First, find that 
has 
and standard deviation 
.
Then find the mean and standard deviation of 
.
First, find that
Then find the mean and standard deviation of
Consider the discrete random variable 
that takes the following values with the corresponding probabilities:
with 
with 
with 
with 
Compute the variance of the distribution.
Consider the discrete random variable
Compute the variance of the distribution.
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The variance of a discrete random variable is computed as
for all the values of 
that the random variable 
can take.
First, we compute 
, which is the expected value. In this case, it is 
.
So we have
The variance of a discrete random variable is computed as
for all the values of
First, we compute
So we have
Clothes 4 Kids uses standard boxes to ship their clothing orders and the mean weight of the clothing packed in the boxes is 
pounds. The standard deviation is 
pounds. The mean weight of the boxes is 
pound with a standard deviation of 
pounds. The mean weight of the plastic packaging is 
pounds per box, with a 
pound standard deviation. What is the standard deviation of the weights of the packed boxes?
Clothes 4 Kids uses standard boxes to ship their clothing orders and the mean weight of the clothing packed in the boxes is
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Note that the weight of a packed box = weight of books + weight of box + weight of packing material used.
It is given that 
.
The calculation of the standard deviation of the weights of the packed boxes is
Note that the weight of a packed box = weight of books + weight of box + weight of packing material used.
It is given that
The calculation of the standard deviation of the weights of the packed boxes is
A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
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Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from because each trial is independent of the others.
Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from