1. Suppose that you flip a fair coin three times.
  1. Write the sample space for this experiment.

    2.  
        {HHH, HTH, HHT, HTT, THH, TTH, THT, TTT}
         
  2. What is the probability of all three flips being heads?
Since there are 8 equally likely possibilities, and only one of them has all three heads, the probability is 1/8.

Another way to do this problem is to draw the appropriate branch of the probability tree, which would look like: .

So the probability is (1/2)·(1/2)·(1/2) = 1/8
  1. Suppose you are given a standard deck of playing cards (4 suits, 13 card values).
  1. If you draw one card at random and then replace that card and draw again, what is the probability that you draw a 3 and a 5 (in either order)?

    2.  
          This is the probability of getting a 3 and then a 5 plus the probability of getting a 5 and then a three.

          P(3 then 5) + P(5 then 3) = (1/13)·(1/13) + (1/13)·(1/13) = 1/169 + 1/169 = 2/169.

  2. If you draw one card at random and then replace that card and draw again, what is the probability that you draw the same card both times?
This one is a bit tricky. The probability of getting, for example, the 5 of clubs, twice in a row is (1/52)·(1/52). So for any particular card, the probability of getting it both times is (1/522). But, there are 52 cards in the deck, so we have to add this probability 52 times (it is just like in part a where we added the probability twice for the two possibilities, only now we have 52 possibilities.). So the final probability is 52·(1/522) = 1/52.
  1. Suppose the events A and B have probabilities P(A) = .6 and P(B) = .5
  1. What is P( not A )?

    2.  
      Put another way, if there is a 60% chance A will happen, there is a 40% chance it will not.
       
  2. If the probability of A and B both happening is .2, what is P(A or B)?
  1. A local club is offering a game of chance with the following rules:


    2. A 6-sided die is rolled. If the number that appears is odd, then you receive $2. If a 2 or 4 appear, then you receive $3. If a 6 appears, then you have to give them $6.

    If they wish to charge you $1.50 each time you play this game, is this a good deal for you?

    Die roll Result Probability
    1, 3, 5 $2.00 1/2
    2, 4 $3.00 1/3
    6 –$6.00 1/6
      So, your expected value for this game is:
      (1/2)(2.00)+(1/3)(3.00)+(1/6)(–6.00) = 1 + 1 – 1 = $1.00
      Since you are being charged $1.50 to play, this is not a good deal for you.
       
  2. In a particular company, there are 100 employees. A quality committee is to be made up of 4 of these employees.
  1. If the committee positions are all the same, how many different committees are possible?

    2.  

     


     
  2. If the committee positions are President, Vice President, Secretary, and Treasurer, how many different committees are possible?

 
  1. The stem-and-leaf-plot below gives the ages of a driver’s school class.
1
 7 7 8 9
2
 2 5 6 6 6 7
3
 1 9
4
 2 4 4 5 8 9 9
5
 1 1 1 2 2 3 3 
6
 5 6 4 5 5 7
7
8
 4
  1. How many people were in this class?

    2.  
          33 people
           
  2. How old was the youngest person in the class?

    3.  
          17
           
  3. How old was the oldest person in the class?
84
  1. Consider the data set
2, 5, 8, 1, 6, 8, 2, 10, 2
  1. Find the mode of this data set.

    2.  
          2
           
  2. Find the mean of this data set.
(2 + 5 + 8 + 1 + 6 + 8 + 2 + 10 + 2) / 9 = 44/9
  1. Consider the data set
9, 12, 16, 19, 22, 26, 29, 34, 38, 40
  1. Find the maximum and minimum of this data set.

    2.  
        maximum = 40, minimum = 9
  2. Find the median and lower and upper quartiles of this data set.

    3.  
        median = (22 + 26) / 2 = 24 (we need to do this because we have an even number of elements)
        lower quartile = 16
        upper quartile = 34

     
  3. Sketch a box-plot of this data set.

        4.  
  4. Assume that a particular exam is graded so that the results are normally distributed. The mean score is 75 and the standard deviation is 8.
  1. What percentage of test takers score above 83 on this test?

    2.  
        For this problem, we use the picture on P. 511
        83 is one s.d. above the mean, so 16% of the people scored above that.
         
  2. What percentage of test takers score above 67 on this test?
67 is one standard deviation below the mean, so 84% of the people scored below that.  
  1. Fill in the blank.
  1. If A and B are mutually exclusive, then P(A and B) is ___.

    2.  
              0
               
  2. If A is an event that is sure to happen, then P(A) is ___.

    3.  
              1
               
  3. A data value that is very far from all other data values is called a(n) ___.
outlier
 
  1. Quick Calculations
  1. What is the standard deviation of the data set 1, 2, 3?

    2.  

     


     
  2. Calculate 4!

    3.  
          4! = 1·2·3·4 = 24
           
  3. If the standard deviation of a data set is 5, what is the variance?
variance = (standard deviation)2 = 52 = 25