Mathematics

Probability Distributions

488 Questions

Probability distributions describe the likelihood of different outcomes in a statistical experiment. Key topics include calculating confidence intervals, understanding Type II errors, and working with the standard normal distribution. These concepts are frequently tested in mathematics and statistics sections of various competitive exams.

Standard normal distributionType II error probabilityConfidence interval interpretationPoisson distributionProbability density function

Probability Distributions Questions

Multiple choice

Which of the following is a discrete probability distribution?

  1. Normal distribution

  2. Binomial distribution

  3. Poisson distribution

  4. Uniform distribution
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

Multiple choice

What is the probability of getting a z-score between -1 and 1 in a standard normal distribution?

  1. 0.3413

  2. 0.6826

  3. 0.9545

  4. 0.9973
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of getting a z-score between -1 and 1 in a standard normal distribution is approximately 0.6826.

Multiple choice

What is the probability of getting a sample mean between 100 and 110 from a population with a mean of 105 and a standard deviation of 10, if a sample of size 25 is taken?

  1. 0.3413

  2. 0.6826

  3. 0.9545

  4. 0.9973
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the central limit theorem, the probability of getting a sample mean between 100 and 110 can be approximated by the probability of getting a z-score between (100 - 105) / (10 / √25) = -1 and (110 - 105) / (10 / √25) = 1 in a standard normal distribution. This probability is approximately 0.6826.

Multiple choice

What is the probability of rejecting a null hypothesis when it is true, also known as a Type I error?

  1. 0.01

  2. 0.05

  3. 0.1

  4. 0.2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In hypothesis testing, the probability of rejecting a null hypothesis when it is true is typically set at 0.05, which is known as the significance level.

Multiple choice

What is the probability of failing to reject a null hypothesis when it is false, also known as a Type II error?

  1. 0.01

  2. 0.05

  3. 0.1

  4. 0.2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In hypothesis testing, the probability of failing to reject a null hypothesis when it is false is known as the Type II error. The probability of a Type II error depends on the significance level, the effect size, and the sample size.

Multiple choice

What is the formula for the law of distribution of conjunction over possibility?

  1. $ \Diamond (\phi \wedge \psi) \leftrightarrow \Diamond \phi \wedge \Diamond \psi $
  2. $ \Diamond (\phi \vee \psi) \leftrightarrow \Diamond \phi \vee \Diamond \psi $
  3. $ \Box (\phi \wedge \psi) \leftrightarrow \Box \phi \wedge \Box \psi $
  4. $ \Box (\phi \vee \psi) \leftrightarrow \Box \phi \vee \Box \psi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The law of distribution of conjunction over possibility states that the possibility of a conjunction is equivalent to the conjunction of the possibilities.

Multiple choice

In the 2019 China Girls' Mathematical Olympiad, what was the probability of randomly selecting a prime number between 1 and 100?

  1. 1/4

  2. 1/5

  3. 1/6

  4. 1/7
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Therefore, the probability of randomly selecting a prime number between 1 and 100 is 25 / 100 = 1/4.

Multiple choice

How do you calculate the expected value of a bet?

  1. EV = (Probability of winning * Payout) - (Probability of losing * Stake)

  2. EV = (Probability of winning * Stake) - (Probability of losing * Payout)

  3. EV = (Probability of winning + Probability of losing) * Stake

  4. EV = (Probability of winning * Payout) + (Probability of losing * Stake)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected value of a bet is calculated using the following formula: EV = (Probability of winning * Payout) - (Probability of losing * Stake). This formula takes into account the probability of winning and losing, as well as the payout and stake.