Test problems

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Test problems discussion
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(1.1) A coin loaded to come up heads 2/3 of the
time, is thrown until a head appears. What is the
probability that the even number of tosses is
necessary? Hints (a) What kind of distribution
should be used? Answer P qp q3 p q5 p
q p ( 1 q2 q4 ) qp/(1-q2)
1/4. (1.2) A committee of 5 is chosen from a
group of 8 men and 4 women. What is the
probability the group contains a majority of
men? Answer A either 3 or 4 women. P(A)
(Binomial8,5 Binomial8,4Binomial4,1)
Binomial8.3 Binomial4,2)/Binomial12,5
0.85 (1.3) A box contains tags numbered 1,2,,
n. Two tags are chosen without replacement. What
is the probability they are consecutive
integers? Hint How many pairs satisfy this
condition Answer 2 (n-1)/Binomialn,2 2
(n-1) (n-2)!/n! 2/n. The total probability is
2/n.
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(1.4) If 20 chips produced by a machine are
defective, find the probability that out of 5
chips chosen at random (a) 1 (b) at least 2
chips will be defective. Solution (a) 50.20.84
0.41 (b) 1 0.85 ltprobability of nonegt -
50.20.84 ltprobability of onegt 1- 0.33 0.41
0.26 1.5 In how many ways can 8 people be
seated at a round table if (a) they can seat
anywhere (b) three of them, Bob, Michael and
Jeff, should not sit together (in other words
all the arrangements where B, M and J sit in any
order in three consecutive chairs must be
excluded). Note Only relative positions are
important. Solution Fix the position of one
person and rearrange all other people relative to
him. There will be (8-1)! 7! 5040 possible
permutations. Fix the position of one person not
belonging to the group of three. Find first the
arrangements keeping B, M and J together.
Consider them a unit. It leaves us with 5 objects
and 5! 120 permutations. There is also 3!
permutations within the group of 3. Thus, the
total number of permutations with M B J sitting
together is 1206 720.The answer is 5040 (the
total number of permutations) 720 (permutations
keeping B, M and J together) 4320.
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2.1 A boy has six coins, each of different
denomination. How many different sums of money
can he form? Answer 26 -1 63 or Binomial6,1
Binomial6,2 ,,Binomial6,6 63. (2.2)
The probability that an entering college student
will graduate is 0.6. Find the probability that
out of 5 students (a) none (b) at least two (c)
at least one will graduate Solution p(none)
p(0) 0.45 0.01. (b) 1 p(0) p(1) 1
p(0) 50.6(0.4)40.91 ( c ) p(ngt0)
0.99. (2.3) A box contains tags numbered 1,2,,
20. Two tags are chosen without replacement. What
is the probability their sum is even? Hint How
many even/odd numbers among the tags? How
can an even number be composed of two even or odd
numbers? Answer EvenSum(Even1,Even2) OR
(Odd1,Odd2) P 1/2(9/19) ½9/19 9/19.
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(2 .4) A committee of 6 is chosen from a group
of 9 men and 5 women. What is the probability
the group contains either four men or four
women. P(A) (Binomial9,4Binomial5,2Binomia
l9,2Binomial5,4)/Binomial14,6
0.48. (2.5) A coin loaded to come up heads ¼ of
the time, is thrown until a head appears. What is
the probability that the odd number of tosses is
necessary? Hints (a) What kind of distribution
should be used? Answer P p q2 p q4 p
p /( 1-q2) ¼ 16/74/7.
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(3.1) A coin loaded to come up heads 1/3 of the
time, is thrown until a head appears. What is the
probability that at least 3 tosses is
necessary? Hints (a) What kind of distribution
should be used? Solution P(ngt2) q2 p q3 p
q4 p p q2/( 1-q) (1/3) (4/9)3 4/9
. And here is the alternative and simpler
solution P (ngt2) 1 p(1)-p(2) 1-1/3 2/9
4/9 (3.2) How many straight lines are
determined by 8 points , no 3 of which are
collinear. Answer Binomial8,2 28
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(3.3) Three integers are selected at random from
the set 1,2,.. 10. What is the probability that
the largest of them is 5?Hint To fit the
condition, there are two groups of objects to
chose from . What are they? What is the
distribution to use? Solution The groups are
5 (1 object) and 1,2,3,4 (4 objects). To find
the probability that one object is chosen from
the first group and one from the second, we use
the hypergeometric distribution (see Lecture
3) P Binomial1,1 Binomial4,2/Binomial10,3
1/20. (3.4) What is the probability of
getting a total of 9 (a) twice (b) at least twice
in 6 tosses of a pair of dice. Answer (a) p(2)
Binomial6,2(1/9)2 (8/9)4 0.116 (b)
p(ngt1) 1 p(0) p(1) 1 (8/9)6 6
(8/9)5(1/9) 0.137.
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(3.5) How many 4 digit numbers can be formed
with the 10 digits 0,1,2,3,9 if (a) repetitions
are allowed (b) the repetitions are not allowed
(c) the last digit must be 0 and the repetitions
are not allowed. Answer (a) The first digit can
be chosen in 9 ways since 0 is not allowed. Other
3 can be any of 10. Answer 9000. (b) 9987
4536 (c) 987 504.