Homework 5 Solutions

Ch. 13: 


4.  4/52 x 3/51 x 2/50 x 1/49 x 4/48 is about 1/3,000,000


8.  (a) On one roll, the chance of getting 3 or more spots is 4/6; the
chance of getting 3 or more spots on 4 rolls is (0.67)^4 (which is
about 0.20 - 20%).

    (b) None of the rolls show 3 or more spots if all show 2 or
fewer, and the chance is (2/6)^4 (which is about 0.012 - 1.2%).

    (c) 100% - 20% = 80%



Ch. 14:


5.  (a) False.  A and B can both happen together, with the chance
1/3 x 1/10.  So they aren't mutually exclusive.

    (b) True.  If A happens, the chance of B drops to 0.  That's an
extreme form of dependence.


11.  (a) 13/52 x 12/51 x 11/50 (about 1%)

     (b) 39/52 x 38/51 x 37/50 (about 41%)

     (c) The chance of getting all diamonds is 1%, see (a).  The
chance of not getting all diamonds is 99%.



Ch. 15:


8. The chance of getting 2 heads among the first 5 tosses is
(5!)/((2!) (3!)) (0.5)^5

The chance of getting 4 heads among the last 5 tosses is
(5!)/((4!) (1!)) (0.5)^2

The first 5 tosses are independent of the last 5.  so the answer we
need to multiply:
(5!)/((2!) (3!)) (0.5)^5 x (5!)/((4!) (1!)) (0.5)^2 = 50/1024

So the answer is about 5%.



Special Review Exercises (pp. 263-268)


6.  False.  These data are cross-sectional, not longitudinal.  The
65-year-old men were born in 1928, and went to school when educational
levels were lower than they are now.  That is one reason their incomes
tend to be lower.  The effect of education on income is confounded
with the effect of age.


8.  (a) True.  List (ii) is obtained by doubling each entry on list
(i), then adding 1.  This change of scale cancels on conversion to
standard units (see section 5.6).

    (b) False.  List (ii) is obtained by doubling each entry on list
(i), changing the sign, then adding 1.  This changes the signs in
standard units.


14.  Solid: (i)
     Dotted: (iii)
     Dashed: (ii)


17.  The 25th percentile on the final is about 0.67 in standard units,
about 73 in original units.  In other words, of the students who
scored around 50 on the midterm, you need to find the percentage
scoring over 73 on the final.  The new average is 60, the new SD is
16, and 73 is 0.81 in standard units.  The answer is given by the area
under the normal curve to the right of 0.81, which is about 21%.


18.  There are 30 possible outcomes, all equally likely.  (You can
list them out, to prove it to yourself.  There are 5 possibilities from the
first box and 6 from the second - 5 x 6 = 30.)

  (a) There are two good outcomes, "2" from box 1 and "5" from box 2,
and vice-versa.  So, the chance is 2/30.  Another way to see it is
that probability of drawing either good outcome is (1/5)(1/6) = 1/30.
Since the two outcomes are mutually exclusive, you can add them to get
2/30.

  (b) 5/30.  There are 5 good outcomes: (1,4), (4,1), (2,5), (5,2),
(1,6).  Also, can see it as 5(1/5)(1/6) = 5/30.  (Since each
combination is equally likely, and all are mutually exclusive.)

  (c) 10/30.  There are 10 good outcomes.  (It's easiest to just list
them for yourself.)  So, the chance is 10/30.  You can see it as
10(1/5)(1/6) = 10/30.  (Since each combination is equally likely, and
all are mutually exclusive.)