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Wize University Statistics Textbook > Probability

Probability Problems

Probability of Dependent Events (Conditional Probability)Previous Section
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Example: Solving for Probability Using Contingency Table

40% of children eat strawberries. Of those that eat strawberries, 80% are vegetarian. Half of those that don’t eat strawberries are vegetarian.

Use the following contingency table to help you solve the questions below:

a) If we randomly select someone, what is the probability that he/she eats strawberries and is vegetarian?
0.32

b) What is the probability we pick someone that’s not vegetarian and doesn’t eat strawberries?

0.30

c) If we randomly select a vegetarian, what is the probability that he/she eats strawberries?

0.320.62=0.516\frac{0.32}{0.62}=0.516

d) If we randomly select someone that doesn’t eat strawberries, what is the probability that he/she is vegetarian?

0.300.60=0.50\frac{0.30}{0.60}=0.50

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Example: Dependent Events

The probability Joe MacNeil will win an Emmy (Event A) is 20%; the probability he will win a Golden Globe (Event B) is 15%. Furthermore, the probability that he will only win an Emmy is also 15%.

Use the following contingency table to help you solve the questions below:


P(A)=0.2P\left(A\right)=0.2
P(B)=0.15P\left(B\right)=0.15
P(A∩Bc)=0.15P\left(A\cap B^c\right)=0.15


(a) What's the probability that he will win both an Emmy and a Golden Globe?

P(A∩B)=0.05P\left(A\cap B\right)=0.05


(b) Are winning an Emmy and winning a Golden Globe independent events?

If the events are independent, then P(A∩B)=P(A)P(B)P\left(A\cap B\right)=P\left(A\right)P\left(B\right)
P(A)P(B)=(0.2)(0.15)=0.03P\left(A\right)P\left(B\right)=\left(0.2\right)\left(0.15\right)=0.03
Since P(A∩B)=0.05 ≠0.03P\left(A\cap B\right)=0.05\ \neq0.03, they are not independent events.


(c) What is the probability that Joe will win an Emmy or a Golden Globe?

P(A∪B)=P(A)+P(B)−P(A∩B)P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)
0.2+0.15−0.05=0.30.2+0.15-0.05=0.3


(d) If he does not win an Emmy, what is the probability that he will win a Golden Globe?

P(B∣Ac)=P(Ac∩B)P(Ac)=0.100.80=0.125\displaystyle{P\left(B\mid A^c\right)=\frac{P\left(A^c\cap B\right)}{P\left(A^c\right)}=\frac{0.10}{0.80}=0.125}
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Example: Probability Distribution Table

The number of defective phones in a sample of 5 varies according to the following probability table:


P(try to find∣given)=P(try to find)P(given)P\left(try\ to\ find\mid given\right)=\frac{P\left(try\ to\ find\right)}{P\left(given\right)}

(a) What is the probability that at least 3 will be defective?
P(X=3)+P(X=4)+P(X=5)=0.12+0.11+0.09=0.320P\left(X=3\right)+P\left(X=4\right)+P\left(X=5\right)=0.12+0.11+0.09=0.320

(b) At least 4 are defective. What is the probability that exactly four are defective?
P(exactly 4)P(at least 4)=0.110.11+0.09=0.110.20=0.55\displaystyle \frac{P\left(exactly\ 4\right)}{P\left(at\ least\ 4\right)}=\frac{0.11}{0.11+0.09}=\frac{0.11}{0.20}=0.55
(c) What is the probability that at most 3 are defective, given that at least 2 are defective?

Key: We already know that at least 2 are defective. So to paraphrase: "Given that at 2 or 3 or 4 or 5 are defective, what's the probability that up to 3 are defective?" This eliminates 0 or 1 being defective.

P(2 or 3)P(2 or more)=0.33+0.120.33+0.12+0.11+0.09=0.450.65=0.6923\displaystyle \frac{P(2\ or\ 3)}{P(2\ or\ more)}=\frac{0.33+0.12}{0.33+0.12+0.11+0.09}=\frac{0.45}{0.65}=0.6923

Given that P(A) = 0.6, P(B) = 0.5, and P(Bc|Ac)= 0.8, fill in this contingency table to help answer the following questions.


Gloria is an opera singer. The probability that she will get a standing ovation is 60%. The probability that she will have a wardrobe malfunction is 20%. The probability that she will only get a standing ovation is 55%.

Fill in this contingency table to help answer the questions that follow.

Extra Practice
Probability
The probability that a first-year student takes geography is 0.40.4. The probability that he takes calculus is 0.60.6, and the probability that he takes neither is 0.30.3. What is the probability this student takes both of these courses?
Probability
A drawer contains 6 identical blue socks, 7 identical red socks, and 12 identical green socks. You randomly select 2 socks from the drawer. What is the probability you get a matching pair if you draw the socks without replacement?
Probability
9% of homes built in 2005 will need some renovations over the next 4 years.
Find the probability that none of five selected homes need renovations.
Probability
An ordinary coin is flipped 11 times. Find the probability that the coin shows heads at most 10 times.
Probability
A drawer contains 6 identical blue socks, 7 identical red socks, and 12 identical green socks. You randomly select 2 socks from the drawer. What is the probability you get a matching pair if you draw the socks without replacement?
Probability
Let E and F be events in a sample space where
P(E)=0.70, P(F)=0.40, P(E∩Fc)=0.45P\left(E\right)=0.70,\ P\left(F\right)=0.40,\ P\left(E\cap F^c\right)=0.45
Determine P(Ec∩Fc)P\left(E^c\cap F^c\right)
Probability
A drawer contains 6 identical blue socks, 7 identical red socks, and 12 identical green socks. You randomly select 2 socks from the drawer. What is the probability you get a matching pair if you draw the socks without replacement?
Probability
The probability that a first-year student takes geography is 0.40.4. The probability that he takes calculus is 0.60.6, and the probability that he takes neither is 0.30.3. What is the probability this student takes both of these courses?
Probability
A teacher surveyed her students and found that 30% play soccer, 35% play baseball, and 25% play both sports. If she randomly selects one of her students, what is the probability that this student plays at least one of these 2 sports?
Probability
The exact number of cookies in a large jar varies from jar to jar according to the probability distribution function (PDF) given by:
a) Find the probability that there are exactly 18 cookies given that there are at least 18 cookies in a jar
b) If in fact there are at most 17 cookies in a jar, what is the probability that there will be at least 16
Probability
The probability that a first year student is enrolled in BMOS is 0.40. 30% of all BMOS students take calculus and 20% of students not in BMOS take calculus.
Probability
Mt. Jumbo is a volcano in Venus that erupts up to 5 times a day1. Here is the data for how many times it erupted based on 180 randomly selected days:
Based on the data collected, what is the probability that no more than 1 or at least 4 volcanoes erupted a day? (round to 3 decimal places)
1 A day in Venus is equivalent to 243 Earth days, if you are curious.
Probability
Julian applied for two scholarships. The probability of him winning each is listed below. Assume the two scholarships are independent.
What is the probability that he wins exactly one of the scholarships?
Probability
If an electrical saw is kept dry, the probability that it fails during the warranty period is 0.05. If the saw gets wet, the probability that it fails during the warranty period is 0.13. If 70% of all electrical saws are kept dry, what is the proportion of electrical saws that will fail during the warranty period?
Probability
A 2018 survey reveals that upon graduation, 65% of students are in debt, 33% are living at home, and 40% of the students who are in debt are living at home. Suppose we select a student at random.
Let A be the event that the student is neither in debt nor living at home.
Let B be the event that the student is in debt but not living at home.
Probability
Trish and Stevie are contestants at a game show. Each will be asked to answer 3 trivia questions.
The probability that Trish answers a question correctly is 76%.
The probability that Stevie answers a question correctly is 45%.
Probability
Trish and Stevie are contestants at a game show. Each will be asked to answer 3 trivia questions.
The probability that Trish answers a question correctly is 76%.
The probability that Stevie answers a question correctly is 45%.
Probability
Trish and Stevie are in a newlywed game show where they have to answer questions about each other. There are 3 rounds so each will be asked 3 questions. The probability that Trish answers a question about Stevie correctly is 76%. The probability that Stevie answers a question about Trish correctly is 45%. They are asked different questions so the answers one provides have no impact on the answers the other provides.
In Round 1, they are each asked one question. What is the probability that at least one of them answers correctly?
Probability
Katrina has 20 black cats, 10 white cats, 5 tuxedo cats, and 15 calico cats. Among all her cats, 30% of them are female.
(a) You randomly pick thee cats with replacement. What is the probability that exactly two of the three are black cats?
(b) You randomly pick thee cats with replacement. What is the probability that all three are the same gender?
Probability
In an urn there are 4 black and 5 white balls. You draw three balls without replacement.
Probability
In an urn there are 4 black and 5 white balls. You draw three balls without replacement.
Determine the probability that
a) you draw 3 black balls
Probability
The probability that a first-year student takes geography is 0.4. The probability that he takes calculus is 0.6, and the probability that he takes neither is 0.3. What is the probability the student is taking both geography and calculus?
Probability
Let A and B be two events in a sample space S where P(𝐴∩𝐵c)=0.5P\left(𝐴∩𝐵^c\right)=0.5, P(𝐵)=0.4P\left(𝐵\right)=0.4, P(𝐴)=0.7P\left(𝐴\right)=0.7
Probability
Mt. Jumbo is a volcano in Venus that erupts up to 5 times a day1. Here is the data for how many times it erupted based on 180 randomly selected days:
Based on the data collected, what is the probability that no more than 1 or at least 4 volcanoes erupted a day? (round to 3 decimal places)
1 A day in Venus is equivalent to 243 Earth days, if you are curious.
Probability
Jean-Marc is a talented violinist - but not that talented - and he's a contestant at a talent show at the Jonquière Music Festival. There are two ways to qualify to the semifinals: to get enough points from the preliminary judges or to obtain the popular vote from the audience. The contestant who wins the popular vote automatically qualifies, regardless of how many points they receive from the judges - they could even be last place!
The probability that Jean-Marc will get enough points from the judges to qualify is 38%. The probability that Jean-Marc will win the popular vote is 15%. If he does not get enough points from the judges, there is a 7% probability that he is saved by winning the popular vote.
What is the probability that he will qualify to the semifinals?
Probability with rolling dice
You roll two fair dice: Die #1 and Die #2. What is the probability of rolling the same number or the sum of the two die is 6?
Probability
Carl needs to buy his wife, Margo, an anniversary gift: necklace, purse, or calculator. The probability that Margo will like the necklace is 70%. She already has a lot of purses so the probability that will like a purse is only 20%. A calculator will upset her big time. Carl will go to only one store to buy a gift. The likelihood of him going to each store is:
Probability
Half of students own an Apple product. One-quarter of all students wear glasses. 35% of all students do not wear glasses and nor own Apple products.
Complete the contingency table. Enter values in two decimal places X.XX (e.g. 0.35).
Probability
Lara took 40 classes at Wize University. She assumed that half of faculty members prefer to be called "Dr. (Name)" and half prefer to be called "Professor (Name)", so she randomly called 20 of them "Dr. (Name)" and 20 of them "Professor (Name)". In reality, only 35% of her instructors prefer to be called "Dr. (Name)" and 65% prefer to be called "Professor (Name)". Furthermore, amongst those she called "Dr.", only 6 of them preferred it. What percent of her instructors did she address by their preferred title?
Probability
P(A) = 0.6
P(B) = 0.5
P(Bc|Ac)= 0.8
Probability
Jasmine did not study very hard and only knows half of the material for her statistics exam. Thankfully, the exam is all multiple-choice with 4 options per question. If she knows the material, she will certainly get the question correct. If she doesn't know the material, she will guess.
She got Question 17 correct. What is the probability that she guessed?
Probability: Contingency table
A marketing company surveyed 500 customers at a mall. They asked if they each customer two questions:
1) Did you see a Jugo Boss ad? YES/NO
2) Did you (i) make a purchase at Jugo Boss, (ii) went to the store but only browsed, or (iii) did not go to the store?
Probability
Jasmine did not study very hard and only knows half of the material for her statistics exam. Thankfully, the exam is all multiple-choice with 4 options per question. If she knows the material, she will certainly get the question correct. If she doesn't know the material, she will guess.
(a) Indeed, she expects to get each question correct with a probability of greater than 50%. Find the exact probability that she responds correctly on a given question.
Probability
P(A)=0.9P\left(A\right)=0.9
P(B)=0.3P\left(B\right)=0.3
Which of the following must be true? [Select the best answer. Click on 'HINT' if you need help.]
Probability
You randomly draw four cards from a deck of 52 cards. What is the probability that you will draw 2 Queens and 2 Hearts?
Probability
You toss two fair die. What is the probability that the difference between the larger value and the smaller value is equal to 2?
Probability
P(A)=0.4P\left(A\right)=0.4
P(Bc)=0.8P\left(B^c\right)=0.8
P(Bc∣Ac)=0.85P\left(B^c\mid A^c\right)=0.85
Probability
There is a 70% chance of rain tomorrow and a 5% chance that the stock market will crash. Assuming that the two events are independent, determine the probability that exactly one of the two events will occur.
Probability of Independent Events
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Probability
Katrina has 20 black cats, 10 white cats, 5 tuxedo cats, and 15 calico cats. Among all her cats, 30% of them are male.
Probability
We asked 100 customers at Louie Baton what they purchased. Here are the results:
30 said they bought accessories (A)
20 said they bought a bag (B)
Probability: Independent Events
Trish and Stevie are in a newlywed game show where they have to answer questions about each other. There are 3 rounds so each will be asked 3 questions. The probability that Trish answers a question about Stevie correctly is 76%. The probability that Stevie answers a question about Trish correctly is 45%. They are asked different questions so the answers one provides have no impact on the answers the other provides.
In Round 1, they are each asked one question. What is the probability that at least one of them answers correctly?
Probability
Trish and Stevie are contestants at a game show. Each will be asked to answer 3 trivia questions.
The probability that Trish answers a question correctly is 76%.
The probability that Stevie answers a question correctly is 45%.
Probability Problems
Trish and Stevie are contestants at a game show. Each will be asked to answer 3 trivia questions.
The probability that Trish answers a question correctly is 76%.
The probability that Stevie answers a question correctly is 45%.
Probability
(v) Suppose it costs $10 to play. The prize is $20. What is the expected net profit from a single play?