Identify the distribution type
A box contains 6 red balls and 4 blue balls. If 3 balls are drawn randomly without replacement, what is the probability that exactly 2 of the balls are red?
Hypergeometric Distribution: HGeom(N=10,M=6,n=3)
P(X=2)=(((6),(2))*((10-6),(3-2)))/(((10),(3)))
A call center operator answers 70% of calls successfully. What is the probability that the operator will answer their 4th successful call on the 9th attempt?
Negative Binomial Distribution: NBin(r=4,p=0.7)
P(X=9)=((9-1),(4-1))*(0.7)^4 *(1-0.7)^(9-4)
A spinner is divided into 8 equal sections. What is the probability that the pointer lands on an even number?
Uniform Distribution
Unif(n=8)
If the probability of a factory worker completing a task in one trial is 0.5, what is the probability that the worker completes the task for the 3rd time on the 10th trial?
Negative Binomial Distribution: NBin(r=3,p=0.5)
P(X=10)=((10-1),(3-1))*(0.5)^3 *(1-0.5)^(10-3)
A bus arrives randomly between 0 and 15 minutes after you arrive at the bus stop. What is the probability that the bus will arrive within the first 10 minutes?
Uniform Distribution: Unif(0,15)
P(X<=10)=int_(0)^10 1/(15-0) dx
A traffic light turns green every 10 minutes on average. What is the probability that the light will turn green exactly 3 times in the next 30 minutes?
Poisson Distribution: Pois(𝜆=30/10=3)
P(X=3)=3^3 /(3!) *e^-3
A car randomly travels between 0 and 100 miles per hour. What is the probability that it travels between 20 and 40 miles per hour?
Uniform Distribution: Unif(0,100)
P(20<=X<=40)=int_(20)^40 1/(100-0) dx
A deck of 52 cards contains 4 aces. If 5 cards are dealt from the deck without replacement, what is the probability that exactly 1 ace is drawn?
Hypergeometric Distribution: HGeom(N=52,M=4,n=5)
P(X=1)=(((4),(1)) *((52-4),(5-1)))/(((52),(5)))
A factory produces light bulbs, and 90% of them are defect-free. If a random sample of 10 light bulbs is selected, what is the probability that exactly 2 bulbs are defective?
Binomial Distribution: Bin(n=10,p=0.1)
P(X=2)=((10),(2))*(0.1)^2 (1-0.1)^(10-2)
The probability of a student answering a multiple-choice question correctly is 0.75. What is the probability that the first correct answer occurs on the 3rd question?
Geometric Distribution: Geom(p=0.75)
P(X=3)=(1-0.75)^(3-1) *(0.75)
A company has a 5% chance of finding a defect in each product produced. What is the probability that the first defective product is found on the 6th trial?
Geometric Distribution: Geom(p=0.05)
P(X=6)=(1-0.05)^(6-1) (0.05)
A factory produces bolts that are normally distributed with a mean length of 10 cm and a standard deviation of 0.2 cm. What is the probability that a randomly selected bolt has a length between 9.8 cm and 10.2 cm?
Normal Distribution: Norm(𝜇=10,𝜎2=(0.2)2=0.04)
Z_1 =(9.8-10)/0.2 =-1, Z_2 =(10.2-10)/0.2 =1
P(9.8<=X<=10.2)=P(Z_2 =1)-P(Z_1 =-1)
P(9.8<=X<=10.2)=0.8413-0.1587=0.6826
A call center receives an average of 4 calls per minute. What is the probability that exactly 6 calls are received in a minute?
Poisson Distribution: Pois(𝜆=4)
P(X=6)=4^6 /(6!) *e^(-4)
A fair die is rolled until a 3 is rolled. What is the probability that the first 3 occurs on the 4th roll?
Geometric Distribution: Geom(p=1/6)
P(X=4)=(1-(1/6))^(4-1) *(1/6)
The heights of adult women in a country are normally distributed with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected woman is taller than 68 inches?
Normal Distribution: Norm(𝜇=65,𝜎2=(3)2=9)
Z=(68-65)/3 =1
P(X>68)=1-P(Z<=1)=1-0.8413=0.1587
In a class of 20 students, each student has a 60% chance of passing an exam. What is the probability that exactly 12 students pass?
Binomial Distribution: Bin(n=20,p=0.6)
P(X=12)=((20),(12))*(0.6)^12 *(1-0.6)^(20-12)
A website experiences an average of 3 visitors per minute. What is the probability that more than 5 minutes will pass before the next visitor arrives?
Exponential Distribution: Exp(𝜆=3)
P(X>5)=e^(-3*5)
If a random variable X follows a particular distribution between 0 and 10, what is the probability that X is less than 5?
Uniform Distribution: Unif(0,10)
[a,b]=[0,10]
P(X<5)=int_(0)^4 1/(10-0) dx
A committee of 10 members is formed from a group of 30 people, of which 10 are women. What is the probability that exactly 4 women are selected?
Hypergeometric Distribution: HGeom(N=30,M=10,n=10)
P(X=4)=(((10),(4))*((30-10),(10-4)))/(((30),(10)))
In a group of 12 students, 4 are from the science department and 8 are from the arts department. If 5 students are randomly selected, what is the probability that exactly 2 are from the science department?
Hypergeometric Distribution: HGeom(N=12,M=4,n=5)
P(X=2)=(((4),(2))*((12-4),(5-2)))/(((12),(5))
The total time it takes to finish three rounds of a game is, on average, 20 minutes. What is the probability that it will take less than 50 minutes to finish three rounds?
Gamma Distribution: Gamma(𝛼=3,𝜆=3/20=0.15)
If the probability of a customer buying a product at a store is 0.1, what is the probability that the first sale occurs on the 7th visit?
Geometric Distribution: Geom(p=0.1)
P(X=7)=(1-0.1)^(7-1) *(0.1)
A lightbulb has a 0.2 chance of burning out each time it is turned on. What is the probability that it takes more than 5 uses before it burns out?
Geometric Distribution: Geom(p=0.2)
P(X>5)=(1-0.2)^5
A basketball player has a 0.6 probability of making a shot. What is the probability that the player makes his 5th shot on the 7th attempt?
Negative Binomial Distribution: NBin(r=5,p=0.6)
P(X=7)=((7-1),(5-1))*(0.6)^5 *(1-0.6)^(7-5)
A company receives an average of 10 customer support calls per hour. What is the probability that it will take more than 10 minutes before the next call comes in?
Exponential Distribution: Exp(𝜆=10)
P(X>1/6)=e^(-10*1/6)
The total waiting time for a customer to be served in a queue is known to have a mean of 30 minutes and a variance of 15 minutes. What is the probability that a randomly selected customer will wait less than 25 minutes?
Normal Distribution: Norm(𝜇=30,𝜎2=15)
Z=(25-30)/(sqrt(15)) =-5/3.87 =-1.29
Z=-1.29=0.0985=9.85%
A store receives an average of 3 customers per minute. What is the probability that exactly 2 customers will arrive in the next minute?
Poisson Distribution: Pois(𝜆=3)
P(X=2)=3^2 /(2!) *e^(-3)
A light bulb has a 1,000-hour average lifespan. What is the probability that a randomly chosen light bulb will last more than 1,200 hours?
Exponential Distribution: Exp(𝜆=1/1000=0.001)
P(X>1200)=e^(-0.001*1200)
A basketball player has a 70% free throw success rate. What is the probability that she makes 8 out of 10 free throws?
Binomial Distribution: Bin(n=10,p=0.7)
P(X=8)=((10),(8)) *(0.7)^8 *(1-0.7)^(10-8)
A hospital emergency room receives 2 patients per hour on average. What is the probability that exactly 5 patients will arrive in the next 3 hours?
Poisson Distribution: Pois(𝜆=2*3=6)
P(X=5)=6^5 /(5!) *e^(-6)
A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?
Binomial Distribution: Bin(n=5,p=0.5)
P(X=3)=((5),(3))*(0.5)^3 *(1-0.5)^(5-3)
A factory produces 100 parts, 15 of which are defective. If a quality control officer randomly inspects 10 parts, what is the probability that exactly 3 defective parts are found?
Hypergeometric Distribution: HGeom(N=100,M=15,n=10)
P(X=3)=(((15),(3))*((100-15),(10-3)))/(((100),(10)))