Answer :
First, let's understand the context and variables involved in this problem.
Johnni is answering 8 multiple-choice questions by guessing. Each question has 4 answer choices, so the probability of guessing any question correctly (p) is:
[tex]\[ p = \frac{1}{4} = 0.25 \][/tex]
We are interested in finding the probability that Johnni gets exactly 3 questions correct out of 8. This situation can be modeled using the binomial probability formula:
[tex]\[ P(k \text{ successes}) = {}_n C_k \cdot p^k \cdot (1-p)^{n-k} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of questions (8 in this case).
- [tex]\( k \)[/tex] is the number of correct answers we are finding the probability for (3 in this case).
- [tex]\( p \)[/tex] is the probability of guessing a question correctly (0.25).
- [tex]\( 1-p \)[/tex] is the probability of guessing a question incorrectly (0.75).
- [tex]\( {}_n C_k \)[/tex] is the binomial coefficient, calculated as:
[tex]\[ {}_n C_k = \frac{n!}{k! \cdot (n-k)!} \][/tex]
For our problem:
[tex]\[ n = 8 \][/tex]
[tex]\[ k = 3 \][/tex]
Now, we'll calculate [tex]\( {}_8 C_3 \)[/tex]:
[tex]\[ {}_8 C_3 = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \][/tex]
[tex]\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 3! = 3 \times 2 \times 1 \][/tex]
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
So:
[tex]\[ {}_8 C_3 = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \][/tex]
Now we can use the binomial formula:
[tex]\[ P(3 \, \text{successes}) = 56 \cdot (0.25)^3 \cdot (0.75)^{8-3} \][/tex]
Calculate [tex]\( (0.25)^3 \)[/tex] and [tex]\( (0.75)^5 \)[/tex]:
[tex]\[ (0.25)^3 = 0.25 \times 0.25 \times 0.25 = 0.015625 \][/tex]
[tex]\[ (0.75)^5 = 0.75 \times 0.75 \times 0.75 \times 0.75 \times 0.75 = 0.2373046875 \][/tex]
Now, multiply these values with the binomial coefficient:
[tex]\[ P(3 \, \text{successes}) = 56 \cdot 0.015625 \cdot 0.2373046875 \][/tex]
[tex]\[ P(3 \, \text{successes}) = 56 \cdot 0.0037060546875 = 0.2076416015625 \][/tex]
Finally, round this result to the nearest thousandth:
[tex]\[ P(3 \, \text{successes}) \approx 0.208 \][/tex]
Therefore, the probability that Johnni got exactly 3 questions correct is approximately [tex]\( 0.208 \)[/tex]. Thus, the correct answer to the question is:
[tex]\[ \boxed{0.208} \][/tex]
Johnni is answering 8 multiple-choice questions by guessing. Each question has 4 answer choices, so the probability of guessing any question correctly (p) is:
[tex]\[ p = \frac{1}{4} = 0.25 \][/tex]
We are interested in finding the probability that Johnni gets exactly 3 questions correct out of 8. This situation can be modeled using the binomial probability formula:
[tex]\[ P(k \text{ successes}) = {}_n C_k \cdot p^k \cdot (1-p)^{n-k} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of questions (8 in this case).
- [tex]\( k \)[/tex] is the number of correct answers we are finding the probability for (3 in this case).
- [tex]\( p \)[/tex] is the probability of guessing a question correctly (0.25).
- [tex]\( 1-p \)[/tex] is the probability of guessing a question incorrectly (0.75).
- [tex]\( {}_n C_k \)[/tex] is the binomial coefficient, calculated as:
[tex]\[ {}_n C_k = \frac{n!}{k! \cdot (n-k)!} \][/tex]
For our problem:
[tex]\[ n = 8 \][/tex]
[tex]\[ k = 3 \][/tex]
Now, we'll calculate [tex]\( {}_8 C_3 \)[/tex]:
[tex]\[ {}_8 C_3 = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \][/tex]
[tex]\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 3! = 3 \times 2 \times 1 \][/tex]
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
So:
[tex]\[ {}_8 C_3 = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \][/tex]
Now we can use the binomial formula:
[tex]\[ P(3 \, \text{successes}) = 56 \cdot (0.25)^3 \cdot (0.75)^{8-3} \][/tex]
Calculate [tex]\( (0.25)^3 \)[/tex] and [tex]\( (0.75)^5 \)[/tex]:
[tex]\[ (0.25)^3 = 0.25 \times 0.25 \times 0.25 = 0.015625 \][/tex]
[tex]\[ (0.75)^5 = 0.75 \times 0.75 \times 0.75 \times 0.75 \times 0.75 = 0.2373046875 \][/tex]
Now, multiply these values with the binomial coefficient:
[tex]\[ P(3 \, \text{successes}) = 56 \cdot 0.015625 \cdot 0.2373046875 \][/tex]
[tex]\[ P(3 \, \text{successes}) = 56 \cdot 0.0037060546875 = 0.2076416015625 \][/tex]
Finally, round this result to the nearest thousandth:
[tex]\[ P(3 \, \text{successes}) \approx 0.208 \][/tex]
Therefore, the probability that Johnni got exactly 3 questions correct is approximately [tex]\( 0.208 \)[/tex]. Thus, the correct answer to the question is:
[tex]\[ \boxed{0.208} \][/tex]
Answer:
Step-by-step explanation:
To find the probability that Johnni got exactly 3 out of 8 questions correct on a multiple-choice quiz, we use the binomial probability formula:
(
successes
)
=
(
)
(
1
−
)
−
P(k successes)=(
k
n
)p
k
(1−p)
n−k
where:
=
8
n=8 (total number of questions),
=
3
k=3 (number of correct answers),
=
1
4
p=
4
1
(probability of getting a question correct),
1
−
=
3
4
1−p=
4
3
(probability of getting a question incorrect).
First, calculate the binomial coefficient
(
8
3
)
(
3
8
):
(
8
3
)
=
8
!
3
!
(
8
−
3
)
!
=
8
!
3
!
⋅
5
!
=
8
×
7
×
6
3
×
2
×
1
=
56
(
3
8
)=
3!(8−3)!
8!
=
3!⋅5!
8!
=
3×2×1
8×7×6
=56
Next, calculate
p
k
and
(
1
−
)
−
(1−p)
n−k
:
=
(
1
4
)
3
=
1
64
p
k
=(
4
1
)
3
=
64
1
(
1
−
)
−
=
(
3
4
)
5
=
243
1024
(1−p)
n−k
=(
4
3
)
5
=
1024
243
Combine these to find the probability:
(
3
correct
)
=
(
8
3
)
×
×
(
1
−
)
−
=
56
×
1
64
×
243
1024
P(3 correct)=(
3
8
)×p
k
×(1−p)
n−k
=56×
64
1
×
1024
243
Calculate the product:
56
×
1
64
=
56
64
=
7
8
56×
64
1
=
64
56
=
8
7
7
8
×
243
1024
=
7
×
243
8
×
1024
=
1701
8192
8
7
×
1024
243
=
8×1024
7×243
=
8192
1701
Convert this to decimal form:
1701
8192
≈
0.208
8192
1701
≈0.208
Thus, the probability that Johnni got exactly 3 questions correct is approximately
0.208
0.208
