LECTURE 8: More Probability (Chapter 14 and 16.4)

A. Overview

Quick Review

1. The chance of something is the percentage of time it is expected to happen
2. Chances are between 0% and 100%
3. The chance of something is 100% minus the opposite thing
4. Multiplying Probabilities: The chance that two or more things will happen equals the chance that the first will happen multiplied by the chance that the second happens and so forth given that a prior event does not preclude the occurrence of a future event. happening.

B. Listing the Ways

For some problems it might help you to list ways and it may help you to understand calculating chances.

If you toss a fair coin, you have two possible outcomes.

The chance of the coin landing heads up is 1/2.

The chance of the coin landing tails up is 1/2.

Let's move to a situation where you are tossing a coin 3 times:

There are 8 possible outcomes (2x2x2):

HHH HHT HTH HTT THH TTH THT TTT

When asked "what is the chance of getting 3 heads?" By listing the outcomes it's easy to see that it's 1/8. And if asked, "what is the chance of not getting 3 heads" it's easy to see that it's 7/8.

But listing outcomes may not be optimal it a situation like this one: you have 5 employees in your store. Employee A shows up 90% of the time, B shows up 75% of the time, C shows up 50% of the time, D shows up 90% of the time, and E shows up 60% of the time. What is the chance that at least one employee will be absent on a given day?

Try listing the ways....(don't)

Instead, you might think about:

Chance at least one absent = 100% - (chance of all 5 not absent)

C. Adding and Multiplying Probabilities

1. Two events A and B are MUTUALLY EXCLUSIVE if the events A and B can't happen together.

   example: I have a deck of cards, I deal a card to you, it can be a heart or it can be a spade, but it can't be both.

   What is the chance of getting dealt a heart? 13/52

   What is the chance of getting dealt a spade? 13/52

2. When asked to find the chance of at least one of two things (or more) things happening, ask if they are mutually exclusive. If they are, you can add the chances.

   example: What is the chance that the card I deal to you is a heart or a spade? 13/52 + 13/52 = 26/52 = 50%

   contrast it with: I deal a card to you. What is the chance that it is a heart? 13/52. I put the card back and deal another card to you, what is the chance that it is a spade? 13/52.

   contrast it with: I deal two cards. I turn over the first card and it is a heart. What is the chance that the second one is a spade? 13/52 * 13/51. (Conditional probabilities)

3. Two events A and B are INDEPENDENT if knowing whether or not A happens does not help in predicting whether or not B happens.

NOTE: under INDEPENDENCE, A & B happen together, they just don't affect each other. Think of rolling dice.

D. Box Models (16.4)

This is just a tool to help you understand Chapter 17.

There is a problem on the practice midterm which uses this tool. I've indicated it on the problem (#10).

Three things to think about:

1. What are the possible outcomes?
2. How many (or what probability or percentage) of each outcome?
3. How many draws do I get?

Example from Chapter 16 p.286, problem 7. The score will be like the sum of 25 draws from a box with tickets that read +4, -1, -1, -1, -1.

E. Homework (due 10/30/98)

• Chapter 14 Exercise Set C: 4 (p. 247)
• Chapter 14 Exercise Set D: 3, 7 (p. 250)