Statistics: Bayes' Theorem and Conditional Probability
Using conditional probability, tables, trees, and Bayes' theorem
Statistics: Bayes' Theorem and Conditional Probability
Using conditional probability, tables, trees, and Bayes' theorem
Statistics - Grade 9-12
- 1
A survey of 200 students found that 120 play a sport, 80 play an instrument, and 45 play both a sport and an instrument. If a student is chosen at random, what is the probability that the student plays an instrument given that the student plays a sport?
Use P(instrument given sport) = P(instrument and sport) divided by P(sport).
The probability is 45/120 = 0.375, or 37.5%. Among the students who play a sport, 45 out of 120 also play an instrument. - 2
A medical test is used to screen for a condition. The condition affects 2% of the population. The test correctly gives a positive result for 95% of people who have the condition, and it gives a false positive result for 8% of people who do not have the condition. If a randomly selected person tests positive, what is the probability that the person actually has the condition?
Compare true positives with all positive test results.
The probability is about 0.195, or 19.5%. Using Bayes' theorem, P(condition given positive) = (0.95)(0.02) / [(0.95)(0.02) + (0.08)(0.98)] = 0.019 / 0.0974, which is about 0.195. - 3
In a school, 55% of students take Spanish, 30% take French, and 12% take both Spanish and French. What is the probability that a randomly selected student takes French given that the student takes Spanish?
The probability is 12/55, which is about 0.218, or 21.8%. This is found by dividing the probability of taking both languages by the probability of taking Spanish. - 4
A factory has two machines. Machine A makes 70% of the products and has a defect rate of 3%. Machine B makes 30% of the products and has a defect rate of 6%. If a randomly selected product is defective, what is the probability that it came from Machine B?
Find the part of all products that are defective and made by each machine.
The probability is about 0.462, or 46.2%. The probability of a defective product from B is (0.30)(0.06) = 0.018. The total probability of a defect is (0.70)(0.03) + (0.30)(0.06) = 0.039. Therefore, P(B given defective) = 0.018/0.039, which is about 0.462. - 5
A bag contains 5 red marbles and 7 blue marbles. One marble is drawn and not replaced. Then a second marble is drawn. What is the probability that the second marble is blue given that the first marble was red?
The probability is 7/11, or about 0.636. After one red marble is removed, there are still 7 blue marbles and 11 total marbles left. - 6
A two-way table shows the results of a class survey about homework completion and quiz success. Of 90 students, 50 completed the homework, 40 did not, 42 completed the homework and passed the quiz, and 18 did not complete the homework but passed the quiz. What is the probability that a student completed the homework given that the student passed the quiz?
The condition is that the student passed the quiz, so use only the students who passed.
The probability is 42/60 = 0.700, or 70.0%. A total of 42 + 18 = 60 students passed the quiz, and 42 of those students completed the homework. - 7
Suppose P(A) = 0.40, P(B) = 0.25, and P(A and B) = 0.10. Find P(A given B) and P(B given A).
P(A given B) = 0.10/0.25 = 0.400. P(B given A) = 0.10/0.40 = 0.250. These conditional probabilities are different because they use different conditions. - 8
A spam filter marks 90% of spam emails as spam and incorrectly marks 4% of regular emails as spam. Suppose 20% of all emails are spam. If an email is marked as spam, what is the probability that it is actually spam?
Do not ignore the base rate of spam emails.
The probability is about 0.849, or 84.9%. Using Bayes' theorem, P(spam given marked) = (0.90)(0.20) / [(0.90)(0.20) + (0.04)(0.80)] = 0.18/0.212, which is about 0.849. - 9
A card is drawn from a standard 52-card deck. Let A be the event that the card is a king, and let B be the event that the card is a face card. What is P(A given B)? Assume face cards are jacks, queens, and kings.
The probability is 4/12 = 1/3, or about 0.333. There are 12 face cards in the deck, and 4 of those face cards are kings. - 10
A city uses two weather models. Model X is used 60% of the time and is correct 80% of the time. Model Y is used 40% of the time and is correct 70% of the time. If a randomly selected forecast was correct, what is the probability that it came from Model X?
Use Bayes' theorem with the event correct as the condition.
The probability is about 0.632, or 63.2%. The probability of a correct forecast from X is (0.60)(0.80) = 0.48. The total probability of a correct forecast is 0.48 + (0.40)(0.70) = 0.76. Therefore, P(X given correct) = 0.48/0.76, which is about 0.632. - 11
In a group of 150 people, 90 drink coffee, 60 drink tea, and 30 drink both coffee and tea. Are the events drinking coffee and drinking tea independent? Explain using probabilities.
For independent events, P(A and B) must equal P(A) times P(B).
The events are not independent. P(coffee) = 90/150 = 0.60 and P(tea) = 60/150 = 0.40, so P(coffee)P(tea) = 0.24. But P(coffee and tea) = 30/150 = 0.20, which is not equal to 0.24. - 12
A rare condition affects 1 in 500 people. A test has a sensitivity of 99%, meaning it is positive for 99% of people with the condition. It has a specificity of 97%, meaning it is negative for 97% of people without the condition. If a person tests positive, what is the probability that the person has the condition?
Specificity of 97% means the false positive rate is 3%.
The probability is about 0.062, or 6.2%. The condition rate is 0.002. The true positive probability is (0.99)(0.002) = 0.00198, and the false positive probability is (0.03)(0.998) = 0.02994. So P(condition given positive) = 0.00198/(0.00198 + 0.02994), which is about 0.062. - 13
A club has 40 members. There are 24 juniors and 16 seniors. Of the juniors, 15 volunteer at an event. Of the seniors, 12 volunteer at the event. If a club member is selected at random and is known to have volunteered, what is the probability that the member is a senior?
The probability is 12/27, or about 0.444. There are 15 + 12 = 27 volunteers total, and 12 of those volunteers are seniors. - 14
A website has visitors from phones and computers. 65% of visitors use phones, and 35% use computers. Of phone users, 8% make a purchase. Of computer users, 12% make a purchase. What is the probability that a randomly selected visitor made a purchase?
Add the purchase probabilities from both device groups.
The probability is 0.094, or 9.4%. Use the law of total probability: P(purchase) = (0.65)(0.08) + (0.35)(0.12) = 0.052 + 0.042 = 0.094. - 15
Using the website information from the previous problem, if a randomly selected visitor made a purchase, what is the probability that the visitor used a computer?
Use Bayes' theorem with purchase as the condition.
The probability is about 0.447, or 44.7%. The probability of a computer user making a purchase is (0.35)(0.12) = 0.042. The total probability of a purchase is 0.094, so P(computer given purchase) = 0.042/0.094, which is about 0.447.