Homework 2

BSTA 550

Due: Thursday October 17, 2024
Modified

October 10, 2024

Directions

Please turn in this homework on Sakai. Please submit your homework in pdf format. You can type your work on your computer or submit a photo of your written work or any other method that can be turned into a pdf. The Adobe Scan phone app is an easy way to scan photos and compile into a PDF. Please let me know if you greatly prefer to submit a physical copy. We can work out another way for you to turn in homework.

Try to complete all of the problems listed below at some point this quarter! You may want to save some of them for studying later! Only turn in the ones listed in the “Turn In” column. Please submit problems in the order they are listed.

You must show all of your work to receive credit.

Chapter Turn In Extra Problems
22* TB # 1 # 3, 5, 7, 25, 27, 30, 31, 39-41, 43-48
3 TB # 10, NTB # 1 # 4, 9, 12, 13**
4 TB # 5 # 1, 4, 11, 13
5 TB # 17, NTB # 2 # 1, 9, 11

* Please note the following for Chapter 22:

  • See the table on pg. 277, which summarizes some key combinatorics concepts.

  • Problems 39-48 are a set that build on one another and more advanced than the other problems. It’ll be much easier to do #42 after doing 39-41.

  • I highly recommend reading Chapter 23, which is a series of case studies in counting: poker hands and Yahtzee.

**For #3.13, mathematically solve for the sample size instead of plugging in numbers and guessing.

Non-textbook problems (NTB)

  1. Recall from class, that we defined events \(A,B,\) and \(C\) to mutually independent if both (1) and (2) below hold. This point of this exercise is to show that \((1)\nRightarrow (2),\) and \((2)\nRightarrow (1).\) \[\begin{array}{cc} (1) & \mathbb{P}(A\cap B\cap C)=\mathbb{P}(A)\mathbb{P(}B)\mathbb{P(}C) \\ (2) & \mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P(}B) \\ & \mathbb{P}(A\cap C)=\mathbb{P}(A)\mathbb{P(}C) \\ & \mathbb{P}(B\cap C)=\mathbb{P}(B)\mathbb{P(}C)% \end{array}%\]

    1. Suppose two different fair dice are rolled. Let events \(A,B,\) and \(C\) be defined in the following way: \[\begin{array}{cl} A: & \text{Roll a total of 7} \\ B: & \text{First die is a 6} \\ C: & \text{Second die is a 2}% \end{array}%\]

      Show that condition \((2)\) holds, but that condition \((1)\) does not.

    2. Suppose two different fair dice are rolled. Let events \(A,B,\) and \(C\) be defined in the following way: \[\begin{array}{cl} A: & \text{Roll a 1 or 2 on the first die} \\ B: & \text{Roll a 3, 4, or 5 on the second die} \\ C: & \text{Roll a total of 4, 11, or 12}% \end{array}%\]

      Show that condition \((1)\) holds, but that condition \((2)\) does not.

  2. A new drug is packaged to contain 30 pills in a bottle. Suppose that 98% of all bottles contain no defective pills, 1.5% contain one defective pill, and 0.5% contain two defective pills. Two pills from a bottle are randomly selected and tested. What is the probability that there are 2 defective pills in the bottle given that one of the two tested pills is defective?