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Two fair dice are rolled. Find the probability that the sum is 2.
A fair die is rolled 2 times. Find the probability of at least one six.
The probability that a flight is late is 0.78. Find the probability that it is not late.
In a class of 35 students, 12 are girls, 12 play soccer and 2 girls play soccer. A student is chosen at random. Find P(girl or soccer player).
Events A and B are mutually exclusive, with P(A) = 0.31 and P(B) = 0.24. Find P(A and B).
A spinner lands on red with probability 0.21 and on blue with probability 0.32. Find P(red or blue).
A screening test was given to 316 people: 47 had the disease and tested positive, 8 had the disease and tested negative, 34 did not have it and tested positive, 227 did not have it and tested negative. Find P(disease | positive).
P(A) = 0.60, P(B) = 0.22 and P(A and B) = 0.5. Find P(A | B).
A test detects a disease in 87% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.87
2. P(disease | positive) = 0.87
3. P(disease and positive) = 0.87
A test detects a disease in 85% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.85
2. P(disease | positive) = 0.85
3. P(disease and positive) = 0.85
A test detects a disease in 82% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.82
2. P(disease | positive) = 0.82
3. P(disease and positive) = 0.82
A test detects a disease in 86% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.86
2. P(disease | positive) = 0.86
3. P(disease and positive) = 0.86
A test detects a disease in 89% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.89
2. P(disease | positive) = 0.89
3. P(disease and positive) = 0.89
A test detects a disease in 97% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.97
2. P(disease | positive) = 0.97
3. P(disease and positive) = 0.97
A test detects a disease in 95% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.95
2. P(disease | positive) = 0.95
3. P(disease and positive) = 0.95
A test detects a disease in 94% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.94
2. P(disease | positive) = 0.94
3. P(disease and positive) = 0.94
A test detects a disease in 93% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.93
2. P(disease | positive) = 0.93
3. P(disease and positive) = 0.93
A test detects a disease in 83% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.83
2. P(disease | positive) = 0.83
3. P(disease and positive) = 0.83
A test detects a disease in 91% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.91
2. P(disease | positive) = 0.91
3. P(disease and positive) = 0.91
A test detects a disease in 88% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.88
2. P(disease | positive) = 0.88
3. P(disease and positive) = 0.88
A test detects a disease in 96% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.96
2. P(disease | positive) = 0.96
3. P(disease and positive) = 0.96
A test detects a disease in 90% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.90
2. P(disease | positive) = 0.90
3. P(disease and positive) = 0.90
A test detects a disease in 81% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.81
2. P(disease | positive) = 0.81
3. P(disease and positive) = 0.81
A test detects a disease in 92% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.92
2. P(disease | positive) = 0.92
3. P(disease and positive) = 0.92
A test detects a disease in 80% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.80
2. P(disease | positive) = 0.80
3. P(disease and positive) = 0.80
A test detects a disease in 98% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.98
2. P(disease | positive) = 0.98
3. P(disease and positive) = 0.98
A test detects a disease in 84% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.84
2. P(disease | positive) = 0.84
3. P(disease and positive) = 0.84
A test detects a disease in 99% of people who have it. Which statement says the same thing?
1. P(positive | disease) = 0.99
2. P(disease | positive) = 0.99
3. P(disease and positive) = 0.99
Two independent machines fail with probabilities 0.8 and 0.7. Find the probability that both fail.
P(A) = 3/10, P(B) = 6/10 and P(A and B) = 18/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 8/10 and P(A and B) = 40/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 8/10 and P(A and B) = 64/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 5/10 and P(A and B) = 40/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 3/10 and P(A and B) = 9/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 8/10 and P(A and B) = 32/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 2/10 and P(A and B) = 4/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 5/10 and P(A and B) = 10/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 4/10 and P(A and B) = 20/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 4/10 and P(A and B) = 8/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 6/10 and P(A and B) = 24/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 7/10 and P(A and B) = 56/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 3/10 and P(A and B) = 15/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 5/10 and P(A and B) = 35/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 7/10 and P(A and B) = 35/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 2/10 and P(A and B) = 6/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 5/10 and P(A and B) = 20/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 2/10 and P(A and B) = 12/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 7/10 and P(A and B) = 21/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 4/10 and P(A and B) = 12/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 7/10 and P(A and B) = 42/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 2/10 and P(A and B) = 8/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 6/10 and P(A and B) = 36/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 8/10 and P(A and B) = 56/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 5/10 and P(A and B) = 30/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 6/10 and P(A and B) = 48/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 8/10 and P(A and B) = 48/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 8/10 and P(A and B) = 16/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 6/10 and P(A and B) = 12/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 4/10 and P(A and B) = 24/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 7/10 and P(A and B) = 49/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 2/10 and P(A and B) = 10/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 6/10 and P(A and B) = 42/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 5/10 and P(A and B) = 25/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 3/10 and P(A and B) = 6/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 4/10 and P(A and B) = 28/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 3/10 and P(A and B) = 12/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 2/10 and P(A and B) = 16/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 2/10 and P(A and B) = 14/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 7/10, P(B) = 3/10 and P(A and B) = 21/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 8/10 and P(A and B) = 24/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 5/10, P(B) = 6/10 and P(A and B) = 30/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 2/10, P(B) = 7/10 and P(A and B) = 14/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 6/10, P(B) = 3/10 and P(A and B) = 18/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 4/10 and P(A and B) = 16/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 3/10 and P(A and B) = 24/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 8/10, P(B) = 4/10 and P(A and B) = 32/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 3/10, P(B) = 5/10 and P(A and B) = 15/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
P(A) = 4/10, P(B) = 7/10 and P(A and B) = 28/100. Are A and B independent?
1. Yes: P(A and B) = P(A)·P(B)
2. No, because P(A and B) > 0
3. It cannot be decided
A bag has 5 red and 3 blue marbles. Two are drawn without replacement. Find P(both red).
12% of a population has a condition. A test is positive for 86% of people with the condition and for 11% of people without it. Find P(positive).
22% of a population has a condition. A test is positive for 81% of people with the condition and for 12% of people without it. Find P(condition | positive).
A disease affects 1 in 1,000 people. A test is 93% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 93% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 99% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 99% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 95% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 95% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 98% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 98% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 97% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 97% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 92% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 92% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 91% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 91% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 96% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 96% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 90% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 90% likely to be sick
3. You are certainly sick
A disease affects 1 in 1,000 people. A test is 94% accurate for both sick and healthy people. Your test is positive. What is most likely?
1. You are still more likely healthy: false positives from the many healthy people outnumber the true positives
2. You are about 94% likely to be sick
3. You are certainly sick
5 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?
In how many ways can a team of 3 students be chosen from 5 students?
Two students are chosen at random from a class of 4 to go on a trip. What is the probability that both Ana and Ben are chosen?
Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 5 students stand in a line for a photo?
2. In how many ways can 2 of the 5 students be chosen for a committee?
3. How many handshakes happen if each of 5 students shakes hands once with every other?
Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 7 students stand in a line for a photo?
2. In how many ways can 2 of the 7 students be chosen for a committee?
3. How many handshakes happen if each of 7 students shakes hands once with every other?
Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 8 students stand in a line for a photo?
2. In how many ways can 2 of the 8 students be chosen for a committee?
3. How many handshakes happen if each of 8 students shakes hands once with every other?
Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 9 students stand in a line for a photo?
2. In how many ways can 2 of the 9 students be chosen for a committee?
3. How many handshakes happen if each of 9 students shakes hands once with every other?
Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 6 students stand in a line for a photo?
2. In how many ways can 2 of the 6 students be chosen for a committee?
3. How many handshakes happen if each of 6 students shakes hands once with every other?

Answers

$\frac{1}{36}$
$\frac{11}{36}$
$\frac{11}{50}$
$\frac{22}{35}$
$0$
$\frac{53}{100}$
$\frac{47}{81}$
$\frac{5}{22}$
P(positive | disease) = 0.87
$\frac{14}{25}$
Yes: P(A and B) = P(A)·P(B)
$\frac{5}{14}$
$\frac{1}{5}$
$\frac{99}{151}$
You are still more likely healthy: false positives from the many healthy people outnumber the true positives
$60$
$10$
$\frac{1}{6}$
In how many ways can 5 students stand in a line for a photo?