Untitled Flashcards Set

Chapter 2: Counting Rules, Probability, and More

1. How many ways can you arrange 5 out of 10 books on a shelf?

   • Formula: Permutation formula P(n,k)P(n, k)P(n,k)

2. How many ways can you choose 4 people from a group of 12?

   • Formula: Combination formula C(n,k)C(n, k)C(n,k)

3. What is the probability of rolling a sum of 8 on two six-sided dice?

   • Formula: Basic probability P(event)=favorable outcomestotal outcomesP(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}P(event)=total outcomesfavorable outcomes​

4. A box contains 6 red balls and 4 green balls. What is the probability of picking exactly 3 red balls without replacement?

   • Formula: Hypergeometric distribution

5. In a class of 30 students, 12 are male. If 4 students are randomly selected, what is the probability of selecting 2 males?

   • Formula: Hypergeometric distribution

6. What is the probability of drawing 2 cards that are both red from a deck of cards without replacement?

   • Formula: Conditional probability using the multiplication rule

7. How many ways can you arrange the letters in the word "STATISTICS"?

   • Formula: Permutation of multiset formula

8. If a bag contains 5 red marbles and 7 blue marbles, what is the probability of drawing 1 red and 1 blue without replacement?

   • Formula: Conditional probability using multiplication rule

9. What is the probability of drawing a King from a deck of 52 cards?

   • Formula: Basic probability P(King)=452P(\text{King}) = \frac{4}{52}P(King)=524​

Chapter 3: Discrete Variables and Distributions

10. In a binomial distribution with n=10n = 10n=10 trials and p=0.3p = 0.3p=0.3, what is the probability of exactly 3 successes?

    • Formula: Binomial probability formula P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

11. The average number of phone calls a customer service agent receives per hour is 5. What is the probability of receiving exactly 2 calls in one hour?

    • Formula: Poisson distribution P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ​

12. In a class of 50 students, 20 are female. If 3 students are selected at random, what is the probability that exactly 2 are female?

    • Formula: Hypergeometric distribution

13. If a fair coin is flipped 8 times, what is the probability of getting exactly 4 heads?

    • Formula: Binomial probability formula

14. A store sells an average of 3 products per hour. What is the probability that the store will sell exactly 5 products in the next hour?

    • Formula: Poisson distribution

15. In a binomial distribution with n=12n = 12n=12 and p=0.6p = 0.6p=0.6, what is the probability of getting at least 7 successes?

    • Formula: Binomial cumulative distribution

16. A researcher wants to know the probability of 4 accidents occurring in a given year. The average is 3 accidents per year. What is the probability of exactly 4 accidents?

    • Formula: Poisson distribution

17. The probability of success in a negative binomial experiment is 0.4. If 5 failures occur before the 3rd success, what is the probability?

    • Formula: Negative binomial distribution

Chapter 4: Continuous Variables and Distributions

18. What is the probability that a value from a normal distribution with μ=100\mu = 100μ=100 and σ=15\sigma = 15σ=15 is less than 85?

    • Formula: Z-score formula and normal distribution P(X<x)P(X < x)P(X<x)

19. Find the probability that a random variable from an exponential distribution with rate parameter λ=3\lambda = 3λ=3 is greater than 2.

    • Formula: Exponential distribution cumulative function P(X>x)=e−λxP(X > x) = e^{-\lambda x}P(X>x)=e−λx

20. The average number of emails received per hour is 4. What is the probability of receiving exactly 3 emails in an hour?

    • Formula: Poisson distribution

21. If the lifetime of a light bulb follows an exponential distribution with λ=0.2\lambda = 0.2λ=0.2, what is the probability it lasts more than 5 hours?

    • Formula: Exponential distribution P(X>x)=e−λxP(X > x) = e^{-\lambda x}P(X>x)=e−λx

22. For a normal distribution with μ=50\mu = 50μ=50 and σ=10\sigma = 10σ=10, what is the probability that a random variable is between 40 and 60?

    • Formula: Z-score formula and normalcdf function (TI-84)

23. What is the probability that a random variable from a normal distribution with μ=60\mu = 60μ=60 and σ=5\sigma = 5σ=5 is greater than 65?

    • Formula: Z-score formula and normal distribution

24. If a random variable follows a normal distribution with a mean of 30 and a standard deviation of 4, find the probability it is less than 28.

    • Formula: Z-score formula and normal distribution

25. What is the expected value for a normal distribution with μ=20\mu = 20μ=20 and σ=2\sigma = 2σ=2?

    • Formula: Expected value for a normal distribution E(X)=μE(X) = \muE(X)=μ

26. Find the variance of an exponential distribution with rate λ=2\lambda = 2λ=2.

    • Formula: Variance of exponential distribution Var(X)=1λ2\text{Var}(X) = \frac{1}{\lambda^2}Var(X)=λ21​

27. What is the probability that a continuous uniform random variable between 10 and 50 is less than 30?

    • Formula: Uniform distribution probability P(X<x)=x−ab−aP(X < x) = \frac{x - a}{b - a}P(X<x)=b−ax−a​

28. The height of a certain species of plant follows a normal distribution with a mean of 5 feet and a standard deviation of 0.5 feet. What is the probability a plant is between 4.5 and 5.5 feet tall?

    • Formula: Z-score and normal distribution

29. In a gamma distribution with shape k=3k = 3k=3 and rate λ=1\lambda = 1λ=1, find the expected value and variance.

    • Formula: Expected value E(X)=kλE(X) = \frac{k}{\lambda}E(X)=λk​ and variance Var(X)=kλ2\text{Var}(X) = \frac{k}{\lambda^2}Var(X)=λ2k​

30. For a normal distribution with μ=50\mu = 50μ=50 and σ=10\sigma = 10σ=10, what is the Z-score for X=60X = 60X=60?

    • Formula: Z-score formula Z=X−μσZ = \frac{X - \mu}{\sigma}Z=σX−μ​

Miscellaneous Practice

31. What is the probability of selecting a red card from a standard deck of cards?

    • Formula: Basic probability

32. A company sells an average of 15 units per day. What is the probability that the company will sell exactly 10 units tomorrow?

    • Formula: Poisson distribution

33. In a deck of 52 cards, what is the probability of drawing a Queen?

    • Formula: Basic probability

34. If a fair coin is flipped 5 times, what is the probability of getting exactly 2 heads?

    • Formula: Binomial probability

35. What is the probability of rolling a sum greater than 10 with two six-sided dice?

    • Formula: Basic probability

36. If 3 marbles are drawn from a jar containing 4 red and 6 blue marbles, what is the probability that 2 are red?

    • Formula: Hypergeometric distribution

37. The probability of getting a "heads" on a fair coin is 0.5. What is the probability of getting exactly 4 heads in 10 flips?

    • Formula: Binomial distribution

38. What is the probability of a randomly selected person being taller than 6 feet, if the heights of people are normally distributed with a mean of 5'8" and a standard deviation of 4 inches?

    • Formula: Z-score formula and normal distribution