Binomial
What are two characteristics of the binomial distribution?
1) There are a fixed number of trials.
2) Each trial results in one of two mutually exclusive outcomes. (success/failure).
3) Outcomes of different trials are independent.
4) The probability that a trial results in success is the same for all trials.
What are two characteristics of the geometric distribution?
1) Each trial results in one of two mutually exclusive outcomes. (success/failure)
2) Outcomes of different trials are independent
3) The probability that a trial results in success is the same for all trials
A company manufactures light bulbs with a mean life of 1500 hours and a standard deviation of 111 hours. What is the minimum life expectancy of the top 2% of the bulbs?
invNorm (area: .98, μ=1500, σ=111, LEFT) = 1727.9661 hours
The scores on a standardized test are normally distributed with a mean of 528 and a standard deviation of 97. What is the probability of a student scoring 528?
P(x=528) = 0
The probability of any exact number occurring is always zero.
An electric circuit is designed so that voltage levels are uniformly distributed between 4.00V and 9.00V. Find the height of the rectangle created by this distribution
Height=1/(9-4)
=1/5 or 0.2
A certain type of seed has a 75% germination rate. A gardener plants 150 of these seeds.
What is the expected number of seeds that will not germinate?
The expected number of seeds that will not germinate is:
n*p = 150 seeds * 0.25 = 37.5 seeds.
A basketball player has a 79% free-throw shooting percentage. The player shoots free throws until they miss one.
What is the expected number of free throws the player will take until they miss?
The expected number of free throws until the player misses is 1/p = 1/0.21 ≈ 4.7619 shots
A company manufactures light bulbs with a mean life of 1550 hours and a standard deviation of 107 hours. What life expectancy does the bottom 34% of light bulbs have?
X=invnorm(area=.34, mean=1550, SD=107,LEFT)
=1505.8664 hours or less
The scores on a standardized test are normally distributed with a mean of 528 and a standard deviation of 97.
What is the probability that a randomly selected student scores less than 528 on a test?
P(X<528)=0.5000 (50% is below the mean)
A coffee dispenser machine fills cups at amounts that are uniformly distributed. Specifically, if I select a small 12 oz coffee, the machine dispenses between 10.5 and 12.5 oz. of coffee, following a uniform distribution.
What is the probability of the amount of coffee dispensed being between 11 and 11.5 ounces?
P(11<X<11.5)=base x height
=.5*.5
=.25
A multiple-choice test has 9 questions. Each question has 5 choices, and only one choice is correct. A student guesses randomly on each question.
What is the standard deviation of the number of questions the student answers correctly?
√(np(1-p))
=√(9 * 0.2 * (1-0.2))
=√(9 * 0.2 * 0.8)
=√(1.44)
=1.2 questions
A seven sided die is rolled until a number greater than 5 is rolled.
What is the standard deviation of the number of rolls needed?
The probability of rolling a number greater than 5 on a 7-sided die is 2/7.
For a geometric distribution, the standard deviation is given by:
Standard deviation = √((1-p)/p²)
In this case, p = 2/7 and 1-p = 5/7
So, the standard deviation is:
√(5/7/(2/7)²) ≈ 2.9580 rolls
What is the Z-score associated with the following scenario:
A company manufactures light bulbs with a mean life of 1500 hours and a standard deviation of 111 hours. What is the maximum life expectancy of the bottom 5% of the bulbs?
Looking at the table, we notice there are two probabilities equidistant to .0500(5%). These probabilities are 0.0495 (Z-score=-1.65) and 0.0505 (Z-score=-1.64).
Therefore the Z-score= (-1.65-1.64)/2
= -1.645
The scores on a standardized test are normally distributed with a mean of 614 and a standard deviation of 55. What is the probability that a randomly selected student scores between 495 and 546 on a test?
normalcdf (lower: 495, upper: 546, μ = 614, σ = 55) = 0.0929
An electric circuit is designed so that voltage levels are uniformly distributed between 5.00V and 9.00V. Find the probability that a randomly selected circuit has a voltage less than 4.95V or greater than 9V
P(X<4.95 or X>9) = 0
This is out of the range so it's impossible
Suppose 77% of people are right-handed. If you randomly select 6 people, what is the probability that exactly 1 of them is left-handed? (round to 4 decimal places)
binompdf (n=6, p=0.23, x =1)=0.3735
A person continues interviewing for jobs until they receive an offer. If the probability of receiving an offer in each interview is 0.29.
What is the probability that he receives his first offer in between 3 and 7 interviews?
P(3 ≤ x ≤ 7) = P(x ≤ 7) - P(x ≤ 2)
= Geometcdf (p=0.29, x=7) - Geometcdf (p=0.29, x=2)
= 0.4131
A company manufactures light bulbs with a mean life of 1550 hours and a standard deviation of 107 hours. What life expectancy does the middle 34% of light bulbs have?
X1 is located 17% to the left of μ=1550 leaving another 33% to the left of X1. This is the area we’ll be inputting in our calculator.
invNorm (area: .33, μ=1550, σ=107, LEFT) = 1502.9293 hours
X2 is located 17% to the right of μ=1550 leaving a total of 67% to the left of X2 (50% below the mean plus 17%). This is the area we’ll be inputting in our calculator.
invNorm (area: .67, μ=1550, σ=107, LEFT) = 1597.0707 hours
Therefore, the life expectancy of the middle 34% of light bulbs is between 1502.9293 and 1597.0707 hours.
What is the Z-score for the following scenario:
The scores on a math test are normally distributed with a mean of 82.7 and a standard deviation of 6.09. What is the probability that a randomly selected student scores above 92?
What is the probability associated with it (using the table)?
Z = (92 - 82.7)/6.09
= 1.53
P(X>92) = 1 - P(X<92)
= 1 - 0.937
= 0.063
Answer one of the following correctly:
How old is Mr. Fernandez?
or
What is Mr. Fernandez' first name?
or
What is Mr. Fernandez' daughter's name?
or
When is Mr. Fernandez' birthday?
40 years old
Irving
Nirvana
August 30th
Write down the factorial and combination form of the following (don't solve)
Suppose 82% of people are right-handed. If you randomly select 6 people, what is the probability that at least 5 of them are right handed?
P(X≥5)=6C5(.82)5(.18)+6C6(.82)6(.18)0
Suppose 72% of people are right-handed. You start asking people, one by one, if they are right-handed.
What is the probability that the fourth person you ask is not the first left-handed person? (round to four decimal places)
P(x≠4)=1- geometpdf (p=.28, x=4)
= 0.8955
A certain type of lightbulb has a lifespan that is approximately normally distributed with a mean of 1000 hours. It is known that 2.5% of the bulbs last less than 850 hours. Find the standard deviation of the lifespan of these bulbs.
Bottom 2.5% → In the Z-Score table we’ll be looking for .0250. The closest value to .0250 is exactly .0250 with a Z-score of -1.96
Z = → -1.96 =(850−1000)/σ
Solve for σ → σ =76.5306
Let X be a random variable whose distribution is normal with mean 29 and standard deviation 2. Which of the following is equivalent to P(x ≥ 27)?
A. P(X ≤ 27)
B. P( 27 < X ≤ 31)
C. P(X < 31)
D. 1 - P(X ≤ 31)
E. P(x ≥ 31)
F. P(X < 27)
G. 1 - P(x ≥ 27)
C. P(X < 31)
P(x ≥ 27)
=normalcdf(lower:27, upper:1E99, mean: 29, SD:2)
=0.8413
P(X < 31)
=normalcdf(lower:-1E99, upper:31,mean: 29, SD:2)
=0.8413
A coffee dispenser machine fills cups at amounts that are uniformly distributed. Specifically, if I select a small 12 oz coffee, the machine dispenses between 10 and 13 oz. of coffee, following a uniform distribution. How large of a coffee cup should I have if I want to be 90% confident that the dispenser won't overfill the cup?
Area = 0.90 = base X height
= 0.90 = base X 1/3
--> Base = 2.7
Cup size = 10 oz + 2.7 oz = 12.7 oz
A multiple-choice test has 9 questions. Each question has 4 choices, and only one choice is correct. A student guesses randomly on each question.
What is the probability that he gets most questions correct? (round to 4 decimal places)
P(X≥5) = 1 - P(X≤4)
= 1 - binomcdf (n=9, p=.25, x=4)
= 0.0489
Two students have a bet. Student A believes she can roll a 6 (six sided die) in 6 rolls or less. If she achieves this, Student B will pay her $10 Fern pesos. Otherwise she has to pay Student B $15 Fern pesos. Calculate the expected value from Student A's perspective.
EV = P(X≤6)($10)+P(X≥7)(-$15)
= geometcdf(p=1/6,x=6)*($10) +
(1-geometcdf(p=1/6,x=6))*(-$15)
= (0.6651)(10)+(.3349)(-15)
= 6.651-5.0235
= $1.6275
A machine is used to fill soda bottles in a factory. The bottles are labeled as containing 2.0 liters, but extra room at the top of the bottle allows for a maximum of 2.25 liters of soda before the bottle overflows. If the machine is set to fill the bottles with an average of 2.08 liters, and standard deviation of 0.15 liter. What is the IQR? (round to 4 decimal places)
Q1
=invnorm(area:.25, mean=2.08, SD=0.15, LEFT)
=1.9788
Q3
=invnorm(area:.75, mean=2.08, SD=0.15, LEFT)
=2.1812
IQR = 2.1812-1.9788 = 0.2024
The scores on a math test are normally distributed with a mean of 77.7 and a standard deviation of 7.77. What is the probability that a randomly selected student scores gets an F (less than 60) or an A (higher than or equal 90)?
P(X<60 or X≥90)= 1-P(60≤X<90)
=1-normalcdf(low:60,upper:90,Mean: 77.7, SD:7.77)
=0.0681
A coffee dispenser machine fills cups at amounts that are uniformly distributed. Specifically, if I select a small 12 oz coffee, the machine dispenses between 9 and 13 oz. of coffee, following a uniform distribution. What is the IQR?
Q1=.25(13-9)=1 -->Q1=9+1=10
Q3=.75(13-9)=3 -->Q3=9+3=12
IQR=13-11=2