The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is ∑ i = 2 12 P ( E i ) . To compute P ( E i ) , define the events E i , n , to be the event that the initial sum is i and the player wins on the nth roll. Argue that P ( E i ) = ∑ n = 1 ∞ P ( E i , n ) .
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps. Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is ∑ i = 2 12 P ( E i ) . To compute P ( E i ) , define the events E i , n , to be the event that the initial sum is i and the player wins on the nth roll. Argue that P ( E i ) = ∑ n = 1 ∞ P ( E i , n ) .
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps.
Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is
∑
i
=
2
12
P
(
E
i
)
. To compute
P
(
E
i
)
, define the events
E
i
,
n
, to be the event that the initial sum is i and the player wins on the nth roll. Argue that
P
(
E
i
)
=
∑
n
=
1
∞
P
(
E
i
,
n
)
.
Answer questions 8.3.3 and 8.3.4 respectively
8.3.4 .WP An article in Medicine and Science in Sports and
Exercise [“Electrostimulation Training Effects on the Physical Performance of Ice Hockey Players” (2005, Vol. 37, pp.
455–460)] considered the use of electromyostimulation (EMS) as
a method to train healthy skeletal muscle. EMS sessions consisted of 30 contractions (4-second duration, 85 Hz) and were carried
out three times per week for 3 weeks on 17 ice hockey players.
The 10-meter skating performance test showed a standard deviation of 0.09 seconds. Construct a 95% confidence interval of the
standard deviation of the skating performance test.
8.6.7 Consider the tire-testing data in Exercise 8.2.3. Compute a 95% tolerance interval on the life of the tires that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these two intervals.
8.6.2 Consider the natural frequency of beams described in
Exercise 8.2.8. Compute a 90% prediction interval on the
diameter of the natural frequency of the next beam of this type
that will be tested. Compare the length of the prediction interval
with the length of the 90% CI on the population mean.
8.6.3 Consider the television tube brightness test described in
Exercise 8.2.7. Compute a 99% prediction interval on the brightness of the next tube tested. Compare the length of the prediction
interval with the length of the 99% CI on the population mean.
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