Classic Probability
A single card is drawn from a standard deck of 52 cards. What is the probability that a face card is drawn?
P(face card) = 12/52 = 0.231
An automobile manufacturer is concerned about a possible recall of its best-selling four-door sedan. If there were a recall, there is a probability of 0.25 of a defect in the brake system, and a probability of 0.18 of a defect in the transmission.
What is the probability that a recall is not caused by a defect in the brake system?
Compliments
P(not brakes) = 1 - P(brakes)
= 1 - 0.25 = 0.75
A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma?
5*3*2 = 30
X P(X)
2 0.08
3 0.23
4 0.05
5 Find P(X=5)
6 0.11
P(X=5) = 1 - 0.08+0.23+0.05+.11
= 1 - 0.47 = 0.53
A charitable organization calls potential donors. Only 10% of potential donors will give over $1000 when asked. What is the probability that a caller working on behalf of the charitable organization will call a total of 50 people and get no more than 6 that donate over $1000?
P(X <= 6) = binomcdf(50, 0.10, 6)
=0.770
True or False: A probability value of 1.23 means that an event has a 123% chance of occurring.
False
Probability can never be more than 1.
A stockbroker has found that the probability a customer will invest in bonds is 0.6, the probability that they will invest in mutual funds is 0.3, and the probability that they will invest in both bonds and mutual funds is 0.15.
What is the probability that a customer will invest in either bonds OR mutual funds?
Addition Rule
P(bonds OR mutual funds)
= P(bonds) + P(mutual funds) - P(both)
= 0.6 + 0.3 - 0.15 = 0.75
How many different 4-digit codes can be created using the numbers 0, 1, 2, 3, 4, 5 if no number can be used more than once, and the last number must be odd?
5*4*3*3 = 180
Let X = price of an item at a clothing store
True or False: X cannot be a discrete random variable because it can take on decimal values.
False
In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, what is the probability that exactly 5 have blowouts?
P(X = 5) = binompdf(15, 0.25, 5)
= 0.165
In a survey of 50 students, each person was asked to state their favorite kind of pizza. 13 said pepperoni, 4 said black olives, 18 said cheese, 9 said sausage, and 6 said mushrooms.
If a single student is selected at random, what is the probability that their favorite pizza is cheese?
P(cheese) = 18/50 = 0.36
Two cards are drawn from a standard 52 card deck, without replacement. Let event A = the first card is a club, event B = the second card is red.
Are these events independent?
No. The cards are drawn without replacement, so the probability of event B will be affected by event A.
A regional spelling been consists of 10 finalists. Determine the number of ways to assign 1st, 2nd, and 3rd place.
10P3 = 10*9*8 = 720
Is the following variable discrete or continuous?
X = the amount of milk produced yearly by a particular cow.
Continuous
Archer can hit the bull’s-eye 48% of the time. Assume that each shot is independent of the previous shots. What is the probability that he shoots 10 arrows and gets at least 7 bull’s-eyes?
P(X >=7) = 1 - P(X <=6)
= 1 - binomcdf(10, 0.48, 6)
= 0.141
A bag of marbles contains 14 purple marbles and 17 pink marbles. When a single marble is drawn from the bag, what is the probability that it is pink?
P(pink) = 17/31 = 0.548
66% of Americans drink coffee every day (according to the National Coffee Association).
If ten Americans are selected at random, what is the probability that all of them drink coffee every day?
Multiplication Rule
(0.66)10 = 0.0157
How many ways are there to select 4 candidates from 13 equally qualified recent graduates for openings in an accounting firm? (assume all openings are for the same position)
13C4 = 715
Find the expected value of the following distribution:
X P(X)
4 0.32
5 0.07
6 0.25
7 0.36
Expected Value = mean = 5.65 = 5.7
A coffee roaster determines that 12% of their 1-pound bags are underfilled. 1-pound coffee bags are shipped 20 to a box. What is the mean and standard deviation for the number of underfilled bags per box?
mean = mu = np = 20(0.12) = 2.4 bags
std. dev. = sigma = sqrt(npq) = sqrt(20*0.12*0.88) = 1.45 bags
Two six-sided dice are rolled. What is the probability that the sum of the dice is 8?
P(sum=8) = 5/36 = 0.139
A bag of marbles contains 14 purple marbles and 17 pink marbles. Two marbles are drawn without replacement. What is the probability that both marbles will be purple?
Multiplication Rule
P(purple and purple) = (14/31)(13/30)
= 0.196
How many different 5-card poker hands contain exactly 3 Queens?
4C3 * 48C2 = 4*1128
= 4512
Find the variance of the following distribution:
X P(X)
4 0.32
5 0.07
6 0.25
7 0.36
variance = (std. dev)2 = (1.259960317)2
= 1.5875 = 1.6
A study estimated that only 93% of drivers have a valid driver’s license. The state highway patrol sets up a road block to check for unlicensed drivers. If they stop 30 drivers, what is the probability that more than 25 of them have a valid driver's license?
P(X > 25) = 1 - P(X <= 25)
= 1 - binomcdf(30, .93, 25)
= 0.945