Statistics 10 Lecture 10 Probability Summarized (Ch 13 & 14)

1. Why do we need probability?

In life, nothing is certain. We like assigning chances (or probability) to everything. Remember, probability is the formal study of the laws of chance and statistics was founded by gamblers…the 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.

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:
  1. 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.