1.           Why do probability and statistics go together?

Remember, probability is the formal study of the laws of chance and statistics was founded by gamblers’Ķthe idea that there is VARIATION in possible outcomes is key to understanding statistics. In later chapters, you will learn that means and other statistics can vary, but in an expected manner subject to the laws of chance.

In statistics, we use the idea of probability only for propositions or events about which we have incomplete knowledge, usually because they lie in the future. If we knew exact details of everything that has ever happened or ever will happen, probability would no longer be necessary or meaningful. So any value we quote for the probability of an event is relative to something we already know or suspect about that event.

2.           Rules Again, slightly different order

  1. Listing the ways (Chapter 14.1) -- it's just a list of possible outcomes. Example, toss two die, there are 36 different outcomes (p. 239)
  2. The chance of an event is the percentage of time it is expected to happen. It is just a number that measures the likelihood of occurrence. Remember fair coins and six-sided die! This means if you could repeat the process again and again and again, in the long run, this outcome would occur the expected percentage of times. An event need not be a single outcome, it could be a couple (e.g. rolling a 1 or a 3 or 5)
  3. In general, probabilities need not be equal, that is coins could be unfair, die could be loaded or employees can be absent 10%, 20% and 50% of the time.
  4. Given that you might be interested in more than one event, you can combine events with some rules:

3.      Rules for Combining Events

  1. Event A and Event B -- both events occur, if they are independent, simply multiply the basic probabilities (die example). If they are not independent, remember to calculate new odds for events (p. 229)
  2. Event A or Event B -- A happens or B happens or they both happen, just not simultaneously. (p. 241) To be MUTUALLY EXCLUSIVE means they have no outcomes in common and one can add their individual probabilities to find the chance that both occur.
  3. Event NOT A = 100% - Event A -- the chance that event A does not happen is 100% minus the chance it does (p. 223). Call this complements, some call it a subtraction rule.

4.           Conditional Probability (p. 226)

               It's a "shrunken" set of outcomes. Die example again. We're going to roll two die one after another. Suppose I am interested in the chance the two die sum to 7. Before I throw the die, I know that 6/36 ways to roll a "7". Now suppose we know that the first die is a "6" what happens to the probability of getting a sum of 7 now? This is a conditional chance, the probability of a sum of 7 given that we have rolled a 6 on the first die.

 

5. Independence (p. 230) and Mutually Exclusive Events

Independence: two events A and B are independent if the occurrence of one has no influence on the probability of the other. Contrast this with mutually exclusive: if A and B are mutually exclusive, if A happens, B cannot happen and if B happens, A cannot happen. So they totally influence the probability of each other.

6. At least once rule

The probability of an event happening at least once in a series of trials is:

1- (probability that A does not happen)# trials

We have already defined a probability so it has the property in which the sum of all the possible outcomes is 1 or 100%. Therefore, when we take the probability of an event not happening we can say it is equal to one minus the probability of the event happening. For example, suppose 20% of all credit card holders default (don’Äôt pay) their bills. We then know that the probability of not picking a credit card holder who defaults is 80%.  The above formula reads, the probability of an event happening at least once is 1 minus the probability of the event not happening raised to the power of #trials.   So for example, if we select 5 credit card holds at random, the chance that at least one will default is 1 ’Äì (.80)5  = 1 - .33 = .67 or 67%

7. In Class Example:  Bullying is Common

Here is the result of a recent study on bullying.  The findings are that 3 in 10 or 30% of students have been bullied or are bullies.

The study had 16,000 respondents (students) let’Äôs assume that the study was well done and representative.

I would like you to think about this headline & article in the context of all the probability we have been learning.

    1. Suppose the study is accurate.  If we were to pick 10 students at random with replacement, how many of the 10 would we expect to be bullies or to have been bullied?
    2. Suppose we were to do it again, that is, pick another 10 students at random with replacement.  How many of the 10 would we expect to be bullies or to have been bullied?
    3. What is the chance that if we picked 10 students at random with replacement none of them would have been bullies or been bullied?
    4. What is the chance that we would find at least 1 in 10 bullied or been a bully?