Exerciseand Solution6.2
What probability should be assigned to the outcome of heads when a biased coin is tossed, ifheads is three timesas likely to come up as tails? What probability should beassigned to the outcome of tails??
Solution: p (T) = ¼, p (H) = ¾.
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Find the probability of each outcome when a biased die isrolled, if rolling a 2 or rolling a 4 is three times as likelyas rolling each of the other four numbers on the die andit is equally likely to roll a 2 or a 4.?
Solution: p(1)=p(3)=p(5)=p(6)= 1/16; p(2)=p(4)=3/8.
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What are the probability ofthese events when we randomlyselect a permutation of {I, 2, 3, 4)?
A)I precede 4.
B)4 precede 1.
C)4 precedes 1 and 4 precedes 2.
D)4 precedes 1, 4 precedes 2, and 4 precedes 3.
E)4 precedes 3 and 2 precedes I.?
Solution:
A)½
B)½
C)¼
D)¼
E)¼
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What is the probability of these events when we randomlyselect a permutation of the 26 lowercase letters of theEnglish alphabet?
A)The permutation consists of the letters in reversealphabetic order.
B)z is the first letter of the permutation.
C)z precedes a in the permutation.
D)a immediately precedes z in the permutation.
E)aimmediately precedes m, which immediately precedesz in the permutation.
F)m, n, and 0 are in their original places in thepermutation.?
Solution:
A)1/26
B)1/26
C)1/2
D)1/26
E)1/650
F)1/15,600
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Find the smallest number of people you need to chooseat random so that the probability that at least two of themwere both born on April 1 exceeds 112?
Solution: 614.
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What is the conditional probability that exactly four headsappear when a fair coin is flipped five times, given thatthe first flip came up heads??
Solution: 1/4.
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What is the conditional probability that a randomly generatedbit string of length four contains at least two consecutiveOs, given that the first bit is a I? (Assume theprobabilities of a 0 and a 1 are the same.)
Solution:3/8.
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Let E and F be the events that a family ofn children haschildren of both sexes and has at most one boy, respectively.Are E and F independent if
A)n = 2?
B)n = 4?
C)n = 5?
Solution:
A)Not independent.
B)Not independent.
C)Not independent.
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A group of six people play the game of "odd person out" to determine who will buy refreshments. Each person flipsa fair coin. If there is a person whose outcome is not thesame as that of any other member of the group, this personhas to buy the refreshments. What is the probability that
there is an odd person out after the coins are flipped once??
Solution:3/16.
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Find the probability that a family with five children doesnot have a boy, if the sexes of children are independentand if
A)a boy and a girl are equally likely.
B)the probability ofa boy is 0.51.
C)the probability that the ith child is a boy is 0.51 -(i/100).
Solution:
A)1/32
B)0.02825
C)0.03795012
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Use pseudo code to write out the probabilistic primalitytest described in Example 16?
Solution:
Procedure probabilistic prime(n, k)
Composite:=false
i :=0
while composite=false and i < k
begin
i:=i+1
choose b uniformly at random with 1 < b < n
apply Miller’s test to base b
if n fails the test then composite:=true
end
if composite=true then print(“composite”)
else print (“probably prime”)
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