Discrete RVs
Usually in the form of a table, this shows the possible values a discrete random variable can have and the probabilities of each.
What is a probability distribution?
This is how a continuous probability distribution is described when all values have the same probability.
Uniform
When two random variables are NOT independent, this is the value that cannot be calculated.
What is the variance/standard deviation.
This is the acronym to help remember what characteristics a Binomial setting must have.
BINS!
Aragon is shooting arrows at a dragon he has come across. From previous experience, he knows he has a hit rate of about 30%. He keeps shooting until he hits the dragon.
Why is this geometric and not binomial?
He is shooting until he hits the dragon, not shooting a set number of times.
The following table shows the probability distribution for how much money is won during a game of roulette. What is the missing probability?
X -$10 $2 $10 $20
P(X=x) .328 .224 .159 ?
.065
If P(X<220) = .587, what is P(X≤220)?
.587, There is no difference between them since there is no defined area under a curve for a single value.
A certain vending machine offers 20 oz bottles of Dr. Pepper for $1.50. If the mean of X, the number of bottles sold, is 50, What is the mean amount of revenue from Dr. Pepper?
$75
The probability of contracting a particularly nasty disease in a hospital is 40%. What is the probability that out of 17 people, 3 people contract the disease?
P(X=3) = 0.0341
Darius Washington Jr. could make free throws with a probability of .72. What is the probability that he would make his first shot on his third throw?
0.0564
Patients were asked to rate their pain on a scale of 1-5 after knee surgery. Let X = the pain score of a randomly selected patient. What is the expected pain rating for a randomly selected patient?
X 1 2 3 4 5
P(X=x) 0.1 0.2 0.3 0.3 0.1
3.1
The torque of a capping machine follows a Normal distribution with a mean of 10 inch-pounds and a standard deviation of 1.2 inch-pounds. How many standard deviations is 1.5 inch-pounds away from the mean?
Z-Score = -7.0833
1.5 inch-pounds is about 7 standard deviations below the mean.
Let W = 0.5X - 1.5
If X has a mean of 32 and a standard deviation of 3.9, Calculate the mean and standard deviation of W.
Mean = 14.5
SD = 1.95
When sampling without replacement, this allows trials to still be considered independent even though they are technically not.
What is the 10% condition?
n ≤ N(.1)
Is the following setting Geometric or not? Explain.
A farmer is randomly selecting tomatoes from a small bin where the probability of selecting a tomato greater than 15 oz is about .05. He keeps selecting tomatoes from the bin until he finds one larger than 15 oz.
Not geometric - the tomatoes are not being replaced, so the selections are not independent of each other.
Let Y denote the number of broken eggs in a randomly selected carton of one dozen eggs at a supermarket. Find the standard deviation of Y.
Y 0 1 2 3 4
P(Y=y) 0.78. 0.11 0.07 0.03 0.01
0.823 eggs
The 2019 Algebra I STAAR test scores are approximately Normal with a mean of 4190 and a standard deviation of 639. What is the probability of a student scoring a 4400 or better?
p=.3712
The resistance of 100-ohm resistors have a mean of 100 ohms and standard deviation of 2.5 ohms while 250-ohm resistors have a mean of 250 ohms and a standard deviation of 2.8 ohms. Calculate the mean and standard deviation of the total resistance of the two components in series for a randomly selected toaster.
mean = 350 ohms
SD = 3.754 ohms
if X~B(30, .27)
Find P(X > 7).
0.5853
Bartholomew owns a lightbulb factory where 3 out of 75 lightbulbs break during the process. Assuming lightbulbs are independent, what is the probability that he randomly selects a broken lightbulb after 5 or more times?
0.8154
Calculate the value of n.
X 5 10 15 20 25
P(X=x) 2n 3n 7n 5n 2n
.0526
The number of people who wash their hands after using the restroom is about 20%. The number of randomly chosen people out of 150 who wash their hands is approximately Normal with a mean of 30 and a standard deviation of 4.899. How many people out of 150 would be considered the 20th percentile?
about 25.877 people out of 150.
ACT scores follow a Normal distribution with a mean score of a randomly selected Senior on the ACT being 24 with a standard deviation of 2. If scores are independent, what is the probability that the difference between in score between two seniors is greater than 5?
SD = 2.828
1-P(-5<x<5) = 0.077
0.0047
Assume that colors of starbursts are independent, so the probability of selecting a red starburst is 0.25. Which event is more likely: Taking 5 or fewer starbursts to find one that is red OR having 5 or fewer red starbursts in a handful of 20?
p1 = 0.7627
p2 = 0.6172
The first event is more likely to happen!