MTH302

1. A basketball player makes free throws with a 70% success rate. What is the probability that their fifth successful free throw happens on their 8th attempt?

a) 0.159 b) 0.354 c) 0.581 d) 0.716

2. In a quality control process, a machine has a 10% defect rate. How many products, on average, will need to be checked until the 5th defective product is found?

a) 5 b) 10 c) 50 d) 100

3. A call center receives calls with a 25% chance of being a complaint. What is the probability that at least 12 calls are needed to get the 4th complaint?

a) 0.7854 b) 0.1837 c) 0.9857 d) 0.3254

4. Let $X$ be a negative binomial random variable representing the number of trials needed to achieve $r$ successes, where the probability of success in each trial is $p$ . Which of the following statements about the moment-generating function (MGF) of the negative binomial distribution is TRUE?

a) The MGF can be used to derive the mean and variance of the distribution. b) The MGF exists for all real values of

t

. c) The first derivative of the MGF at

t = 0

gives the variance of the distribution. d) The MGF simplifies to the MGF of the binomial distribution when

r = 1

.

5. If the probability that a person believes a rumor is 0.6, what is the probability that the 10th person to hear the rumor will be the third person to believe it?

a) 0.028 b) 0.027 c) 0.27 d) 0.28

6. Select the correct statement regarding the mean and variance of the Negative Binomial distribution:

a) Mean is greater than variance. b) Mean is less than variance. c) Mean is equal to variance. d) Mean and variance both are equal to 1.

7. A light bulb manufacturing factory finds that 3 out of every 60 light bulbs are defective. What is the probability that the first defective bulb will be found when the 6th one is tested?

a) 0.0368 b) 0.0386 c) 0.386 d) 0.0863

8. In a geometric distribution, what does the random variable $X$ typically represent?

a) The number of successes before the first failure b) The number of failures before the first success c) The total number of trials in a fixed sample d) The probability of success in a single trial

9. If the probability of passing a pilot exam is 0.7, what is the probability that a student will pass the test on the third try?

a) 0.063 b) 0.63 c) 0.43 d) None of the above

10. A certain area in the United States is, on average, hit by 6 hurricanes per year. Find the probability that in a given year, this area will be hit by fewer than 4 hurricanes.

a) 0.666 b) 0.1515 c) 0.55 d) 0.9

11. A book has 520 pages and contains 390 typographical errors. Assuming a Poisson law for the number of errors per page, find the probability that a randomly chosen page contains no errors.

a) 0.472 b) 0.945 c) 0.999 d) 0.222

12. A factory produces bulbs, and each has a 5% probability of being defective. If a sample of 10 bulbs is taken, what is the probability that exactly 1 bulb is defective?

a) 0.4013 b) 0.3874 c) 0.4500 d) 0.3150

13. If the number of emails received per minute follows a Poisson distribution with a mean of 4, what is the probability of receiving more than 2 emails in a minute?

a) 0.2381 b) 0.7619 c) 0.4752 d) 0.3678

14. What is the first moment (mean) of a Poisson distribution using MGF (Moment Generating Function)?

a)

λ

b)

λ^{2}

c)

e^{λ}

d)

1 - λ

MCQ on Negative Binomial, Geometric, and Poisson Distributions

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