Chapter 6: Probability

 


6.1  The Idea of Probability (pp. 330-333)

1.      In statistics, what is meant by the term random?   We call phenomenon random if individual outcomes are uncertain but there in nonetheless a regular distribution of outcomes in a large number of repetitions.  “Random” in statistics is not a synonym for “haphazard” but a description of a kind of order that emerges only in the long run. 

2.      In statistics, what is meant by probability?    The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.  That is, probability is a long-term relative frequency.

3.      What is probability theory?   Probability theory is the branch of mathematics that describes random behavior.

4.   In statistics, what is meant by an independent trial?    Trials are independent if the outcome
      of one trial does not influence the outcome of any other.

6.2    Probability Models (pp. 335-356)

1.   In statistics, what is a sample space?   The sample space S of a random phenomena is the set
      of all possible outcomes.

2.   In statistics, what is an event?   An event is any outcome or a set of outcomes of a random
      phenomenon.  That is, an event is subset of the sample space.

3.   What is a probability model?   A probability model is a mathematical description of a random
       phenomenon consisting of two parts:  a sample space S and a way of assigning probabilities
       to events.

4.   When counting the number of events in a sample space, what is meant by the multiplication
      principle? 
If you can do one task in a number of ways and a second task in b number of
      ways, then both tasks can be done in a x b number of ways.

5.   What is the difference between sampling with replacement and sampling without
      replacement?   For example, if you are selecting random digits by drawing numbered slips of
      paper from a hat, and you want all ten digits to be equally likely to be selected each draw,
      then after you draw a digit and record it, you must put it back into the hat.  Then the second
      draw will be exactly like the first.  This is referred to as sampling with replacement.  If you
      do not replace the slips you draw, however, there are only nine choices for the second slip
      picked, and eight for the third.  This is called sampling without replacement.

6.   Explain why the probability of any event is a number between 0 and 1.   An event with  
      probability 0 never occurs.  An event with probability 1 occurs on every trial.  The
      probability of any event must be between these two possible outcomes.

7.   What is the sum of the probabilities of all possible outcomes in a sample space?   Because
      some outcome must occur on every trial, the sum of the probabilities for all possible
      outcomes must be exactly 1.

8.   Describe the probability that an event does not occur.   The probability that an event does not
      occur is 1 minus the probability that the event does occur.  The probability that an event
      occurs and the probability that it does not occur always add to 100% or 1.

9.   What is meant by the complement of an event?   The complement of any event A is the event that A does not occur, written as . The complement rule states that .

10.  When are two events considered disjoint?  Two events A and B are disjoint (also called
       mutually exclusive) if they have no outcomes in common and so can never occur
       simultaneously. 

11.  What is the probability of two disjoint events?   If A and B are disjoint.  Then
          P(A or B) = P(A) + P(B).  This is the addition rule for disjoint events.

12.   How do you find the probability of equally likely events in a sample space?   If a random
        phenomenon has k possible outcomes, all equally likely, then each individual outcome has
        probability 1/k.  The probability of any event A is
                    

13   .Explain why the probability of getting heads when flipping a coin is 50%.                      P(heads)
14.   What is the Multiplication Rule for independent events?   Two events A and B are
        independent if knowing that one occurs does not change the probability that the other
        occurs.  If A and B are independent, P(A and B) = P(A)P(B).

15.   Can disjoint events be independent?   No, disjoint events cannot be independent.  If A and B
        are disjoint, then the fact that A occurs tells us that B cannot occur.

16.   If two events A and B are independent, what must be true about Ac and Bc?  If A and B are
        independent then Ac and Bc are also independent and  Ac is independent of B.

6.3           General Probability Rules (pp. 359-379)
1.   What is meant by the union of two or more events?  Draw a diagram.  The union of any
       collection of events is the event that at least one of the collection occurs.

2.   State the addition rule for disjoint events.   If events A, B, and C are disjoint in the sense that
        no two have any outcomes in common, then  P(one or more of A, B, C) = P(A) + P(B) + P(C).

3.   State the general addition rule for unions of two events.   For any two events A and B,
                  P(A or B) = P(A) + P(B) – P(A and B).

4.   Explain the difference between the rules in #2 and #3.   If A and B are disjoint, the event
{A and B} that both occur has not outcomes in it.  This empty event  is the complement of the sample space S and must have probability 0.  So the general addition rule #3 includes #2, the addition rule for disjoint events.

5.   What is meant by joint probability?  The simultaneous occurrence of two events is called a   
       joint event.  The probability of a joint event is called a joint probability. 

6.   What is meant by conditional probability?   The conditional probability P(B|A) is the
      probability that event B occurs given that event A has already occurred.

7.   State the general multiplication rule.   The probability that both of two events A and B happen
      together can be found by P(A and B) = P(A)P(B|A)

8.   How is the general multiplication rule different than the multiplication rule for independent
      events?   If A and B are independent, then P(B|A) = P(B) so the two rules are the same.  If A
      and B are not independent, the rule must be adjusted for this condition.

9.   State the formula for finding conditional probability.   When P(A) > 0, the conditional
      probability of B given A is:         P(B|A) .

10. What is meant by the intersection of two or more events?  Draw a diagram.   The intersection
      of any collection of events is the event that all of the events occur.

11. Explain the difference between the union and the intersection of two or more events.   The
      union is the event that any of them occur, the intersection is the event that they all occur.

12.  State the formula used to determine if two events are independent.   Two events A and B that
       both have positive probability are independent if  P(B|A) = P(B).