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6.1
Long trials of this experiment often approach 40% heads. One theory attributes this surpris-
ing result to a “bottle-cap effect” due to an unequal rim on the penny. We don’t know. But a teach-
ing assistant claims to have spent a profitable evening at a party betting on spinning coins after
learning of the effect.
6.2
The theoretical probabilities are, in order: 1/16, 4/16
1/4, 6/16
3/8, 4/16
1/4, 1/16.
6.3
(b) In our simulation, Shaq hit 52% of his shots. (c) The longest sequence of misses in our run
was 6 and the longest sequence of hits was 9. Of course, results will vary.
6.4
(a) 0. (b) 1. (c) 0.01. (d) 0.6 (or 0.99, but “more often than not” is a rather weak description of an
event with probability 0.99!)
6.5
There are 21 0s among the first 200 digits; the proportion is
0.105.
6.6
(a) We expect probability 1/2 (for the first flip, or for
any
flip of the coin). (b) The theoretical
probability that the first head appears on an odd-numbered toss of a fair coin is
. Most answers should be between about 0.47 and 0.87.
6.7
Obviously, results will vary with the type of thumbtack used. If you try this experiment, note
that although it is commonly done when flipping coins, we do not recommend throwing the tack
in the air, catching it, and slapping it down on the back of your other hand.
6.8
In the long run, of a large number of hands of five cards, about 2% (one out of 50) will con-
tain a three of a kind. (Note: This probability is actually
0.02113.)
6.9
The study looked at regular season games, which included games against poorer teams, and it is
reasonable to believe that the 63% figure is inflated because of these weaker opponents. In the World
Series, the two teams will (presumably) be nearly the best, and home game wins will not be so easy.
6.10
(a) With
n
20, nearly all answers will be 0.40 or greater. With
n
80, nearly all answers will
be between 0.58 and 0.88. With
n
320, nearly all answers will be between 0.66 and 0.80.
6.11 (a)
S
{germinates, fails to grow}. (b) If measured in weeks, for example,
S
{0, 1, 2,
. . .}.
(c)
S
{A, B, C, D, F}. (d) Using Y for “yes (shot made)” and N for “no (shot missed),”
S
{YYYY,
NNNN, YYYN, NNNY, YYNY, NNYN, YNYY, NYNN, NYYY, YNNN, YYNN, NNYY, YNYN,
NYNY, YNNY, NYYN}. (There are 16 items in the sample space.) (e)
S
{0, 1, 2, 3, 4}.
6.12 (a)
S
{all numbers between 0 and 24}. (b)
S
{0, 1, 2,
. . . .
11000}. (c)
S
{0, 1, 2, . . .,
12}. (d)
S
{all numbers greater than or equal to 0}, or
S
{0, 0.01, 0.02, 0.03, . . .}. (e)
S
{all
positive and negative numbers}. Note that the rats can lose weight.
88
4165
1
2
1
1
2
2
3
1
1
2
2
5
...
2
3
21
200
6
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