Probability

Description: This quiz is designed to test your understanding of probability, a branch of mathematics that deals with the likelihood of events occurring.
Number of Questions: 15
Created by:
Tags: probability random variables conditional probability bayes' theorem
Attempted 0/15 Correct 0 Score 0

A fair coin is tossed twice. What is the probability of getting two heads?

  1. 1/2

  2. 1/4

  3. 1/8

  4. 1/16


Correct Option: B
Explanation:

Since the coin is fair, the probability of getting heads on each toss is 1/2. The probability of getting two heads is the product of these probabilities, which is (1/2) * (1/2) = 1/4.

A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of randomly selecting a blue ball?

  1. 5/10

  2. 3/10

  3. 2/10

  4. 1/10


Correct Option: B
Explanation:

The probability of selecting a blue ball is the number of blue balls divided by the total number of balls. So, the probability is 3 / (5 + 3 + 2) = 3/10.

A random variable X has a probability distribution given by P(X = x) = k * x^2, where k is a constant and x = 1, 2, 3. Find the value of k.

  1. 1/6

  2. 1/3

  3. 1/2

  4. 2/3


Correct Option: A
Explanation:

The sum of probabilities for all possible values of X must be equal to 1. So, we have: P(X = 1) + P(X = 2) + P(X = 3) = k * (1^2) + k * (2^2) + k * (3^2) = k * (1 + 4 + 9) = k * 14 = 1. Therefore, k = 1/14.

Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.6, what is the probability of both A and B occurring?

  1. 0.24

  2. 0.16

  3. 0.12

  4. 0.08


Correct Option: A
Explanation:

Since A and B are independent, the probability of both occurring is the product of their individual probabilities. So, P(A and B) = P(A) * P(B) = 0.4 * 0.6 = 0.24.

A box contains 10 defective items and 90 non-defective items. If two items are randomly selected without replacement, what is the probability that both items are defective?

  1. 1/100

  2. 1/50

  3. 1/25

  4. 1/10


Correct Option: A
Explanation:

The probability of selecting a defective item on the first draw is 10/100. After the first draw, there are 9 defective items and 89 non-defective items remaining. The probability of selecting a defective item on the second draw is 9/99. Therefore, the probability of selecting two defective items is (10/100) * (9/99) = 1/100.

A survey found that 60% of people prefer coffee, 30% prefer tea, and 10% prefer both coffee and tea. What is the probability that a randomly selected person prefers coffee or tea?

  1. 0.8

  2. 0.9

  3. 0.7

  4. 0.6


Correct Option: B
Explanation:

The probability of preferring coffee or tea is the sum of the probabilities of preferring coffee and preferring tea, minus the probability of preferring both. So, P(coffee or tea) = P(coffee) + P(tea) - P(coffee and tea) = 0.6 + 0.3 - 0.1 = 0.9.

A continuous random variable X has a probability density function given by f(x) = 2x for 0 ≤ x ≤ 1. Find the probability that X is less than or equal to 0.5.

  1. 0.25

  2. 0.5

  3. 0.75

  4. 1


Correct Option: B
Explanation:

The probability that X is less than or equal to 0.5 is given by the integral of f(x) from 0 to 0.5: P(X ≤ 0.5) = ∫[0, 0.5] 2x dx = [x^2]_[0, 0.5] = 0.5^2 - 0^2 = 0.25.

A random variable X follows a normal distribution with mean μ = 10 and standard deviation σ = 2. What is the probability that X is between 6 and 14?

  1. 0.6827

  2. 0.9545

  3. 0.3173

  4. 0.0455


Correct Option: A
Explanation:

The probability that X is between 6 and 14 is given by the area under the normal distribution curve between these values. We can use the standard normal distribution (Z) to calculate this probability: P(6 ≤ X ≤ 14) = P((6 - 10)/2 ≤ Z ≤ (14 - 10)/2) = P(-2 ≤ Z ≤ 2) = 0.6827.

A company claims that their new product will last for at least 100 hours. A consumer group tests the product and finds that the average lifetime is 95 hours with a standard deviation of 10 hours. Assuming the lifetime follows a normal distribution, what is the probability that a randomly selected product will last for more than 100 hours?

  1. 0.1587

  2. 0.8413

  3. 0.3413

  4. 0.6587


Correct Option: A
Explanation:

We can use the standard normal distribution (Z) to calculate this probability: P(X > 100) = P((X - 95)/10 > (100 - 95)/10) = P(Z > 0.5) = 1 - P(Z ≤ 0.5) = 1 - 0.6915 = 0.1587.

A bag contains 4 red balls, 3 blue balls, and 2 green balls. If two balls are randomly selected without replacement, what is the probability that the first ball is red and the second ball is blue?

  1. 3/14

  2. 6/35

  3. 2/21

  4. 1/10


Correct Option:
Explanation:

The probability of selecting a red ball on the first draw is 4/9. After the first draw, there are 3 red balls, 3 blue balls, and 2 green balls remaining. The probability of selecting a blue ball on the second draw is 3/8. Therefore, the probability of selecting a red ball on the first draw and a blue ball on the second draw is (4/9) * (3/8) = 3/35.

A fair six-sided die is rolled twice. What is the probability of getting a sum of 7?

  1. 1/6

  2. 1/12

  3. 1/18

  4. 1/24


Correct Option: A
Explanation:

There are 36 possible outcomes when rolling a fair six-sided die twice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). So, the probability of getting a sum of 7 is 6/36 = 1/6.

A survey found that 40% of people prefer coffee, 30% prefer tea, and 20% prefer both coffee and tea. What is the probability that a randomly selected person prefers only coffee?

  1. 0.2

  2. 0.3

  3. 0.4

  4. 0.5


Correct Option: A
Explanation:

The probability of preferring only coffee is the probability of preferring coffee minus the probability of preferring both coffee and tea. So, P(coffee only) = P(coffee) - P(coffee and tea) = 0.4 - 0.2 = 0.2.

A random variable X has a probability mass function given by P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6. Find the expected value of X.

  1. 3.5

  2. 4

  3. 4.5

  4. 5


Correct Option: A
Explanation:

The expected value of X is given by E(X) = Σ[x * P(X = x)] for all possible values of X. So, E(X) = 1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6) = 3.5.

A continuous random variable X has a probability density function given by f(x) = 3x^2 for 0 ≤ x ≤ 1. Find the variance of X.

  1. 0.25

  2. 0.5

  3. 0.75

  4. 1


Correct Option: A
Explanation:

The variance of X is given by Var(X) = E(X^2) - [E(X)]^2. First, we find E(X^2): E(X^2) = ∫[0, 1] 3x^2 * x dx = ∫[0, 1] 3x^3 dx = [3x^4/4]_[0, 1] = 3/4. Then, we find [E(X)]^2: [E(X)]^2 = (3.5)^2 = 12.25. Therefore, Var(X) = 3/4 - 12.25 = 0.25.

A company claims that their new product will last for at least 100 hours. A consumer group tests the product and finds that the average lifetime is 95 hours with a standard deviation of 10 hours. Assuming the lifetime follows a normal distribution, what is the probability that a randomly selected product will last for less than 80 hours?

  1. 0.0228

  2. 0.1587

  3. 0.3413

  4. 0.8413


Correct Option: A
Explanation:

We can use the standard normal distribution (Z) to calculate this probability: P(X < 80) = P((X - 95)/10 < (80 - 95)/10) = P(Z < -1.5) = 0.0228.

- Hide questions