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

What is the probability that a system will fail before a specified time (t), given that it has already survived up to time (t_0)?

  1. Reliability

  2. Availability

  3. Conditional Failure Probability

  4. Mean Time to Failure
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The conditional failure probability, denoted by (F(t|t_0)), represents the probability of failure in the interval ((t_0, t]) given that the system has survived up to time (t_0).

Multiple choice

What is the probability that a system will fail between two specified times (t_1) and (t_2)?

  1. Reliability

  2. Availability

  3. Failure Rate

  4. Conditional Failure Probability
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The conditional failure probability, denoted by (F(t_2|t_1)), represents the probability of failure in the interval ((t_1, t_2]) given that the system has survived up to time (t_1).

Multiple choice

What is the probability that a system will be available at a given time (t)?

  1. Reliability

  2. Availability

  3. Failure Rate

  4. Mean Time to Failure
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Availability, denoted by (A(t)), represents the probability that a system will be in a working condition and able to perform its intended function at time (t).

Multiple choice

In a binomial distribution, the probability of success on each trial is denoted by:

  1. p

  2. q

  3. n

  4. r
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a binomial distribution, the probability of success on each trial is denoted by p.

Multiple choice

The probability of failure on each trial in a binomial distribution is denoted by:

  1. p

  2. q

  3. n

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

The probability of failure on each trial in a binomial distribution is denoted by q.

Multiple choice

The number of trials in a binomial distribution is denoted by:

  1. p

  2. q

  3. n

  4. r
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The number of trials in a binomial distribution is denoted by n.

Multiple choice

The probability of obtaining exactly (k) successes in (n) trials in a binomial distribution is given by:

  1. \(P(X = k) = \binom{n}{k} p^k q^{n-k}\)
  2. \(P(X = k) = \binom{n}{k} p^{n-k} q^k\)
  3. \(P(X = k) = \binom{n}{k} p^k\)
  4. \(P(X = k) = \binom{n}{k} q^k\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining exactly (k) successes in (n) trials in a binomial distribution is given by (P(X = k) = \binom{n}{k} p^k q^{n-k}).

Multiple choice

The variance of a binomial distribution is given by:

  1. \(np\)
  2. \(nq\)
  3. \(np(1-p)\)
  4. \(npq\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The variance of a binomial distribution is given by (npq).

Multiple choice

A multiple-choice test has 10 questions, each with 4 possible answers. If a student guesses on each question, what is the probability of getting exactly 5 correct answers?

  1. 0.004

  2. 0.005

  3. 0.006

  4. 0.007
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The probability of getting exactly 5 correct answers in 10 questions is (P(X = 5) = \binom{10}{5} (1/4)^5 (3/4)^5 = 0.006).

Multiple choice

A company receives 100 orders per day. If the probability of a defective order is 0.05, what is the probability of receiving exactly 5 defective orders in a day?

  1. 0.074

  2. 0.084

  3. 0.094

  4. 0.104
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of receiving exactly 5 defective orders in a day is (P(X = 5) = \binom{100}{5} (0.05)^5 (0.95)^95 = 0.084).

Multiple choice

A binomial distribution has a mean of 10 and a standard deviation of 3. What is the probability of obtaining a value between 5 and 15?

  1. 0.683

  2. 0.693

  3. 0.703

  4. 0.713
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of obtaining a value between 5 and 15 is (P(5 \le X \le 15) = P(\frac{5-10}{3} \le \frac{X-10}{3} \le \frac{15-10}{3}) = P(-1.67 \le Z \le 1.67) = 0.693).

Multiple choice

A binomial distribution has a mean of 20 and a standard deviation of 4. What is the probability of obtaining a value greater than 25?

  1. 0.023

  2. 0.033

  3. 0.043

  4. 0.053
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining a value greater than 25 is (P(X > 25) = P(\frac{X-20}{4} > \frac{25-20}{4}) = P(Z > 1.25) = 0.023).

Multiple choice

A binomial distribution has a mean of 50 and a standard deviation of 10. What is the probability of obtaining a value less than 30?

  1. 0.004

  2. 0.014

  3. 0.024

  4. 0.034
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining a value less than 30 is (P(X < 30) = P(\frac{X-50}{10} < \frac{30-50}{10}) = P(Z < -2) = 0.004).

Multiple choice

A binomial distribution has a mean of 100 and a standard deviation of 15. What is the probability of obtaining a value between 70 and 130?

  1. 0.683

  2. 0.693

  3. 0.703

  4. 0.713
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining a value between 70 and 130 is (P(70 \le X \le 130) = P(\frac{70-100}{15} \le \frac{X-100}{15} \le \frac{130-100}{15}) = P(-2 \le Z \le 2) = 0.683).

Multiple choice

A binomial distribution has a mean of 200 and a standard deviation of 20. What is the probability of obtaining a value greater than 220?

  1. 0.023

  2. 0.033

  3. 0.043

  4. 0.053
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining a value greater than 220 is (P(X > 220) = P(\frac{X-200}{20} > \frac{220-200}{20}) = P(Z > 1) = 0.023).