IE 227 RECITATION - 7
15.11.2013
5.5 An archer shoots an arrow into a square target 6 feet on a side whose center we call the
origin. The outcome of this random experiment is the point in the target hit by the arrow. The
archer scores 10 points if she hits bull’s eye – a disk of radius 1 foot centered st the origin; she
scores 5 points if she hits the ring with inner radius 1 foot and outer radius 2 feet centered at
the origin; and she scores 0 points otherwise. For one arrow shot, let S be the score.
a) Determine the value of S for each possible outcome of the random experiment.
b) Is S a discrete random variable? Explain your answer.
Identify in words and as a subset of the sample space each of the following events,
c) {𝑆 = 5}
Answer:
d) {𝑆 > 0}
e) {𝑆 ≤ 7}
f) {5 < 𝑆 ≤ 15}
g) {𝑆 < 15}
h) {𝑆 < 0}
a)
Outcome (x,y)
√𝒙𝟐 + 𝒚𝟐 ≤ 𝟏
Value of S, S(x,y)
10
𝟏 < √𝒙𝟐 + 𝒚𝟐 ≤ 𝟐
5
√𝒙𝟐 + 𝒚𝟐 > 𝟐
0
b)…
5.8 Six men and five women apply for a job at Alpha Inc. Three of the applicants are selected
for interviews. Let X denotes the number of women in the interview pool.
a) What are the possible values of the random variable X?
b) Obtain the number of interview pools corresponding to each possible value of X; that is,
for each possible value x of X, find the number of ways that event {𝑋 = 𝑥} can occur.
c) Describe the event {𝑋 ≤ 1} in words and find the number of ways in which it can occur.
Answer: X: denotes the the number of women in the interview pool.
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a) X={0, 1, 2, 3} Since 3 applicants will be selected.
b) {X=0}, event that there is no women in the pool.
{X=0} = (50). (63)
{X=1}=…
5.9 Consider the random experiment of tossing a coin three times and observing the result (a
head or tail) for each toss. Let Y denotes the difference between the number of heads obtained
and the number of tails obtained in the three tosses of the coin.
a) Construct a table showing the value of Y for each possible outcome.
b) Identify in words and as a subset of the sample space the event {𝑌 = 0}.
Answer: a)
Outcome (w)
Y(w)
(HHH)
3
(HHT)
1
(HTH)
1
(HTT)
-1
(THH)
1
(THT)
-1
(TTH)
-1
(TTT)
-3
5.24 Suppose that two balanced dice are rolled. Let X be the sum of the two faces showing.
a)
Obtain a PMF of the random variable X ?
b)
Construct a probability histogram for X ?
Answer: a) Outcomes (i.e.) : (1,1), (1,2), (2,1), (2,2) …, number of all outcomes = 36
Outcome
X(x)
(1,1)
2
(1,2) (2,1)
3
Probability, 𝑷𝒙 (𝒙)
𝑃𝑥 (𝑥)=probability mass function
2
x
𝑷𝒙 (𝒙)
Outcome
2
1/36
(1,1)
3
2/26
(1,2) (2,1)
4
3/36
(1,3)(3,1)(2,2)
…
b)…
5.42 According to Daily Racing Form, the probability is about 0.67 that the favorite in a horse
race will finish in the money (first, second, or third place). In the next five races, what is the
probability that the favorite finishes in the money
a) Exactly twice
b) exactly four times
c) at least four times
d) Between two and four times, inclusive
e) Determine the probability distribution of the random variable X, the number of times
the favorite finishes in the money in the next five races.
f) Identify the prob. distribution of X as right-skewed, symmetric, or left skewed without
consulting its probability distribution or drawing its probability histogram.
g) Draw a probability histogram for X.
Answer: a) X= number of times the favorite finishes in the money in the next five races
5
𝑃(𝑥 = 2) = ( ) . (0.67)2 . (1 − 0.67)5−2 = 0.1613
2
5
b) 𝑃(𝑥 = 4) = (4). (0.67)4 . (1 − 0.67)5−4 = 0.3325
c) …
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