====== Question 3 Exercise 6.4 ======
Solutions of Question 3 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

=====Question 3(a)=====
A true or false test contains eight questions. If a student guesses the answer for each question, find the probability that $8$ answers are correct.
====Solution====
We have $8$ questions, each question has two options.

Therefore, The state space contains $2^8$ distinct outcomes selected without bias. 
Thus $$n(S)=256$$
Thus the probability for each individual outcome to occur is $$\dfrac{1}{256}$$
$8$ answers are correct.

Let $$A=\{8\}$$
Obviously only one outcome corresponds to this event,

because we can select all question to be correct by one way

i.e. $${ }^8 C_8=\dfrac{8 !}{(8-8) ! 8 !}=1$$
Therefore probability to $8$ answers are correct is:
$$P(A)=\dfrac{1}{256}$$


=====Question 3(b)=====
A true or false test contains eight questions. If a student guesses the answer for each question, find the probability that $7$ answers are correct.
====Solution====
We have $8$ questions, each question has two options.

Therefore, The state space contains $2^8$ distinct outcomes selected without bias. 
$$n(S)=256$$
Thus the probability for each individual outcome to occur is $$\dfrac{1}{256}$$
$7$ answers are correct

Let $$B=\{7\}$$
then possible outcome or to select $7$ answers correct out of $8$ are:
$$n(B)={ }^8 C_7=\dfrac{8 !}{(8-7) ! 7 !}=8$$
Thus the probability that $7$ answers out of $8$ are correct is:
$$P(B)=\dfrac{n(B)}{n(S)}=\dfrac{8}{256}=\dfrac{1}{32}$$


=====Question 3(c)=====
A true or false test contains eight questions. If a student guesses the answer for each question, find the probability that $6$ answers are correct.
====Solution====
We have $8$ questions, each question has two options.

Therefore, The state space contains $2^8$ distinct outcomes selected without bias. 
$$n(S)=256$$
Thus the probability for each individual outcome to occur is $\dfrac{1}{256}$

$6$ answers are correct.

Let $$C=\{6\}$$
now the ways to select $6$ out of $8$ questions are:
$$n(C)={ }^8 C_6=\dfrac{8 !}{(8-6) ! 6 !}=28$$
Thus the probability that $6$ answers are correct out of $8$ is:
$$P(C)=\dfrac{n(C)}{n(S)}=\dfrac{28}{256}=\dfrac{7}{64}$$


=====Question 3(d)=====
A true or false test contains eight questions. If a student guesses the answer for each question, find the probability that at least $6$ answers are correct.
====Solution====
We have $8$ questions, each question has two options.

Therefore, The state space contains $2^8$ distinct outcomes selected without bias. 
$$n(S)=256$$
Thus the probability for each individual outcome to occur is $\dfrac{1}{256}$

At least $6$ answers are correct.

Let $$D=\{6\}$$
Here $6$ question to be correct are at least maximum may be eight correct.

so the possible oulcome in this case
\begin{align}
 n(D)&={ }^8 C_6+{ }^8 C_7+{ }^8 C_8 \\
& =28+8+1=37 . \quad \text { Thus } \\
P(D)-\dfrac{n(D)}{n(D)}&=\dfrac{37}{256}\end{align}





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