Unformatted text preview: ELEC210 Spring 2011 Homework1Solution
1. A random experiment consists of selecting two balls in succession from an urn containing
two black balls and one white ball.
(a) Specify the sample space for this experiment.
(b) Suppose that the experiment is modified so that the ball is immediately put back into
the urn after the first selection. What is the sample space now?
(c) What is the relative frequency of the outcome (white, white) in a large number of
repetitions of the experiment in part a? In part b?
(d) Does the outcome of the second draw from the urn depend in any way on the
outcome of the first draw in either of these experiments?
Solution (16’):
(a) In the first draw the outcome can be black (b) or white (w). If the first draw is black, then
the second outcome can be b or w. However if the first draw is white, then the urn only
contains black balls, so the second outcome must be b. Therefore
(b) In this case, all outcomes can be b or w. Therefore
(c) In part a, the outcome “ww” cannot occur, so .
. . In part b, let N be a large number of repetitions of the experiment. The number of times that the first outcome is “w” is
approximately N/3, since the urn has one white ball and two black balls. Of these N/3
outcomes, approximately 1/3 are also white in the second draw. Thus N/9 of the outcome
is “ww”, and thus . (d) In the first experiment, the outcome of the first draw affects the probability of the
outcomes in the second draw. In the second experiment, the outcome of the first draw
does not affect the probability of the outcomes in the second draw.
2. Three friends (Al, Bob and Chris) put their names in a hat and each draws a name from
the hat. (Assume Al picks first, then Bob, then Chris.)
(a) Find the sample space.
(b) Find the sets A, B, and C that correspond to the events “Al draws his name,” “Bob
draws his name,” and “Chris draws his name.”
(c) Find the set corresponding to the event, “no one draws his own name.”
(d) Find the set corresponding to the event, “everyone draws his own name.”
(e) Find the set corresponding to the event, “one or more draws his own name.”
Solution (15’):
(a) , here “abc” means “Al draws Al, Bob draws Bob, Chris draws Chris”, “acb” means “Al draws Al, Bob draws Chris, Chris draws Bob”
(b) A={“Al draws his name”}={abc, acb}
B={“Bob draws his name”}={abc, cba}
C={“Chris draws his name”}={abc, bac}
(c)
(d)
(e)
3. Find the probabilities of the following events in terms of , and : (a) A occurs and B does not occur; B occurs and A does not occur.
(b) Exactly one of A or B occurs.
(c) Neither A nor B occur.
Solution (16’):
We can get the results easily through using the venn diagram. (a) ,
. (b)
(c)
4. A number x is selected at random in the interval [1, 1]. Numbers from the subinterval [0,
1] occur twice as frequently as those from [ 1, 0]. Let events
. Find the probabilities of A, B and C. Solution (18’):
Since numbers from the subinterval [0, 1] occur twice as frequently as those from [ 1, 0], and
x can only be in [1, 1], we have
, so . ;
, as x can only in [1, 1], then
; , so . 5. A lot of 50 items has 40 good items and 10 bad items.
(a) Suppose we test five samples from the lot, with replacement. Let X be the number of
defective items in the sample. Find . (b) Suppose we test five samples from the lot, without replacement. Let Y be the number
of defective items in the sample. Find . Solution (16’):
(a)
(b) s 6. Find the probability that in a class of 28 students exactly 4 were born in each of the seven
days of the week.
Solution (19’):
For Monday, we pick 4 students from the total 28 students, the number of possible ways is
. For Tuesday, we pick 4 students from the remaining 284=24 students, the number of
possible ways is , Then the total number of possible ways that “exactly four were born in each of the seven days of the week” is And the size of the sample space is , so the probability is . , ...
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 Spring '11
 Prof.ShenghuiSong
 Probability, Probability theory, Outcome, Chris

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