Download ECE-316 Tutorial for the week of June 1-5

ECE-316 Tutorial for the week of June 1-5
Problem 35 – Page 176: refer to lecture notes – part 2, slides 8, 15
A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same
color, then you win $1.10; if they are different colors, then you win -$1.00. (That is, you lose $1.00.)
Calculate
a) The expected value of the amount you win
b) The variance of the amount you win.
Solution:
Define X as the random variable of the amount which you win after withdrawing.
1.10
1
1.10
1
1
1.10
∑
a)
1.10
1
0.0667
1
1.0933
b)Var
∑
Var
1.10
1.0933
1.0889
0.0667
Problem 38 – Page 176: refer to lecture notes – part 2, slides 12, 15, 17
1 and Var
If
a)
2
b) Var 4
5, find
;
3 .
Solution:
a)
Var
Var
2
5
4
4
1
6
4
4
4
4
6
14
Another Solution:
Var 2
2
2
Var
5
b) Var 4
3
2
2
14
1
9Var
9 5
45
Problem 44 – Page 176: refer to lecture notes – part 2, slide 19
A satellite system consists of n components and functions on any given day if at least k of the n
components function on that day. On a rainy day each of the components independently functions with
probability , whereas on a dry day, they each independently function with probability . If the
probability of rain tomorrow is , what is the probability that the satellite system will function?
Solution:
Æ The satellite system may function either tomorrow is rainy or dry.
Æ Whether it will rain or not, the system will function for i= k or k+1 or ….. n components out of n
components function on this day, thus we have for each i a total number of combinations equal to
|
Æ Probability that the satellite will function in a rainy day,
|
∑
1
Æ Probability that the satellite will function in a dry day,
|
is expressed as:
∑
|
is expressed as:
1
Æ Probability that the satellite system will function
|
∑
|
1
∑
1
1
Problem 46 – Page 176: refer to lecture notes – part 2, slide 19
Suppose that it takes at least 9 votes from a 12 member jury to convict a defendant. Suppose also that the
probability that a juror votes a guilty person innocent person is 0.2 whereas the probability that a juror
votes an innocent person as guilty is 0.1. If each juror acts independently and if 65 percent of the
defendants are guilty, find the probability that the jury renders a correct decision. What percentage of
defendants is convicted?
Solution:
Let C denotes to that the jury renders a correct decision.
Let G denotes when defendants are guilty.
Let CV denotes when defendants are convicted.
The probability that the jury renders a correct decision
|
|
12
∑
0.8
0.65
0.2
12
∑
The percentage of defendants is convicted
|
∑
:
0.1
0.35
0.9
:
|
12
0.8
0.2
0.65
∑
12
0.1
0.9
0.35
Problem 47 – Page 176: refer to lecture notes – part 2, slide 19
In some military courts, 9 judges are appointed. However, both the prosecution and defense attorneys are
entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is
not replaced. A defendant in declared guilty if the majority of judges cast votes of guilty, and he or she is
declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will
(independently) vote guilty with probability 0.7, whereas when the defendant is, in fact, innocent, this
probability drops to 0.3.
a) What is the probability that a guilty defendant is declared guilty when there are (i) 9, (ii) 8 and
(iii) 7 judges?
b) Repeat part a) for an innocent defendant.
c) If the prosecution attorney does not exercise the right to a peremptory challenge of a judge, and if
the defense is limited to at most two such challenges, how many challenges should the defense
attorney make if he or she is 60 percent that the client is guilty?
Solution:
Let G denotes to the event of being guilty and D is the event when the defendant is declared guilty.
a) The probability that a guilty defendant is declared guilty
∑
9
0.7
0.3
0.901
|
, when there are 9 judges
when there are 8 judges = ∑
8
when there are 7 judges = ∑
7
0.7
0.3
0.806
0.7
0.3
0.874
b) The probability that an innocent defendant is declared guilty
∑
9
0.3
0.7
|
when there are 9 judges =
0.099
when there are 8 judges = ∑
8
when there are 7 judges = ∑
7
0.3
0.7
0.058
0.3
0.7
0.126
c) We need to find the number of peremptory challenges should the defense attorney make which
minimize the probability of being declared guilty. So we will calculate this probability for all possible
number of peremptory challenges, 0, 1, 2 which corresponds to 9, 8, 7 judges respectively.
Case 1, (9 judges):
|
0.901 0.6
|
0.099 0.4
0.58
Case 2, (8 judges):
|
0.806 0.6
|
0.058 0.4
0.507
Case 3, (7 judges):
|
0.874 0.6
|
0.126 0.4
0.575
Since case 2 results in minimum probability of being declared guilty, then the defense attorney should
make 1 peremptory challenge.
Problem 49 – Page 177: refer to lecture notes – part 2, slide 19
When coin 1 is flipped, it lands on head with probability 0.4; when coin 2 is flipped, it lands on heads
with probability 0.7. One of these coins is randomly chosen and flipped 10 times.
a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
b) Given that the first of these ten flips lands heads, what is the conditional probability that exactly 7
of the 10 flips land on heads?
a) Let C1 denotes when coin 1 is chosen and C2 denotes when coin 2 is chosen.
The probability that the coin lands on heads on exactly 7 of the 10 flips is
| 1
| 2
1
:
2
Under the assumption that coins chosen is equally likely,
10
7
0.4
0.6
10
7
0.5
0.7
0.3
0.5
b) When the first flip is head, then the probability is computed from choosing 6 head flips out of 9 flips in
9
9
both coins case
0.5
0.5 .
0.4 0.6
0.7 0.3
6
6
Let F denotes to the event when the first flip is head.
|
| 1
9
6
1
| 2
| 1
1
0.4
0.6
0.5
2
0.4 0.5
| 2
9
6
0.7
0.7 0.5
2
0.3
0.5
0.55