University:
Kentucky State University
Course:
MAT 321 | Probability/Statistics I
Academic year:

2024
Views:

246

Pages:

12
Author:

embohescay9l

Question 1 (4 marks) In how many different ways can the letters of the word "COMPUTATION" be arranged if: The vowels must always come together? Answer: There are 10 letters in the word "COMPUTATION." We can treat the 4 vowels (O, U, A, I) as a single unit. So, we have 7 units to arrange: 6 consonants and 1 unit of vowels. The number of ways to arrange these 7 units is 7! = 5040. Now, the 4 vowels can be arranged among themselves in 4! = 24 ways. Therefore, the total number of arrangements is 5040 * 24 = 120960. Question 2 (15 marks) How many different car plate numbers can we make if it consists of 3 letters and 5 numbers? Answer: For the letters, we have 26 choices for the first letter, 26 choices for the second letter, and 26 choices for the third letter. So, 1 the total number of ways to choose 3 letters is 26 * 26 * 26 = 17576. For the numbers, we have 10 choices for each digit (0-9). So, the total number of ways to choose 5 numbers is 10 * 10 * 10 * 10 * 10 = 100000. Therefore, the total number of different car plate numbers is 17576 * 100000 = 1757600000. Question 3 (4 marks) A basketball player typically makes 70% of his foul shots. And we assume that the attempts are independent. Find the probability that: a. He misses for the first time on his fifth attempt. b. He makes his first basket on his fifth attempt. Answer: a. The probability of missing a shot is 1 - 0.7 = 0.3. The probability of missing the first four shots and then making the fifth is: 0.3 * 0.3 * 0.3 * 0.3 * 0.7 = 0.00567. b. The probability of making the first basket on the fifth attempt is: 0.3 * 0.3 * 0.3 * 0.3 * 0.7 = 0.00567. **Q4. (7 Marks)** **Short answers only.** 1. Suppose you have a distribution X with mean 3 and standard deviation 8. Define a new random variable Y = 4X - 3. Find the mean and standard deviation of Y. E[Y] = E[4X - 3] = 4E[X] - 3 = 43 - 3 = 9 Var(Y) = Var(4X - 3) = 4²Var(X) = 168² = 1024 SD(Y) = √Var(Y) = √1024 = 32 2. A researcher randomly selects 4 fish from a tank, and puts each of the 4 selected fish into different containers. How many ways can this be done? 4! = 432*1 = 24 3. An experimenter is randomly sampling 5 objects in order from among 49 objects. What is the total number of samples in the sample space? P(49, 5) = 49! / (49-5)! = 49! / 44! = 228326080 4. A person eating at a cafeteria must choose 3 of the 14 vegetables on offer. Calculate the number of elements in the sample space for this experiment. C(14, 3) = 14! / (3! * (14-3)!) = 364 5. How many license plates can be made using 3 digits and 3 letters if repeated digits and letters are not allowed? 10 * 9 * 8 * 26 * 25 * 24 = 11232000

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